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Sustainable Growth in a North-South Trade Model
Copyright © IFAC Modeling and Control of Economic Systems, Klagenfurt. Austria. 2001
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SUSTAINABLE GROWTH IN A NORTH-SOUTH TRADE MODEL Francisco Cabo • , Guiomar Martin-Hernin •• , Maria Pilar Martinez-Garcia •••
• Dept. Econom(a Aplicada (Matemdticas). Avda. Valle Esgueva, 6, Valladolid 47011, Spain •• Dept. Econom(a Aplicada (Matemdticas). Avda. Valle Esgueva, 6, Valladolid 4701 1, Spain ••• Dept. Metodos Cuantitativos para la Econom(a. Campus del Espinardo, Murcia 30100, Spain
Abstract: A model with two different regions that trade with each other is presented. The South specializes in a resource intensive good while the North specializes in a capital intensive good. The North-South trade and the management of natural resources are modelled in a dynamic way so that sustainable economic growth can be analyzed . The existence of balanced paths that allow sustained growth in the North and permanent growth of consumption in the South without exhausting Southern natural resources is proved. The transitional dynamics to a balanced path is also studied. Copyright © 2001 IFAC Keywords : Differential games, economic systems, dynamic models, numerical simulation, steady states
only two different kinds. The stocks of these resources present different dynamic evolutions. The South harvests from these two resources and with the extracted amount it produces a resource intensive good. The North buys this good on the international market and uses it as an intermediate "natural" good to produce a capital intensive good.
I. INTRODUCTION
Most of the authors that study endogenous growth and the environment focus on an isolated country without taking into account trade relationships with other countries. Even those that consider several countries and settle a differential game do assume that all countries are identical, see (Hettich, 2000).
Within the endogenous growth literature, technological knowledge either augments labor productivity in the final output production function or expands product variety or raises product quality, see (Barro and Sala-i-Martin, 1995). On the contrary, in the presented paper technological knowledge enhances the productivity of an intermediate "natural" good. Following Bovenberg and Smulders (1995), an environmental R&D sector devoted to increasing the efficiency of the intermediate "natural" good in the North's output production is assumed.
In contrast to the previous works and following (Cabo et al., 1999) a model with two different regions that trade with each other is presented. Within this framework of North-South trade and environmental problems, following the static approach by Chichilnisky (1994) and Cope land and Taylor (1994), the South specializes in a resource intensive good. Nevertheless the North-South trade and the management of natural resources is modelled in a dynamic way so that sustainable economic growth can be analyzed. All possible renewable resources in the South are simplified to
75
Romer (1986) eliminates the diminishing returns to the factors, assuming that knowledge is a by-product of capital accumulation. A firm that increases its physical capital learns at the same time how to produce more efficiently. This is known as the learning by doing effect. Additionally, it is assumed that each firm's knowledge is a public good which spills over instantly across all other firms at no cost. This makes labor productivity dependent not on each firm's capital stock but on the whole economy's. These two combined effects are considered to specify how labor affects the North's production function.
efficiency in the use of the intermediate "natural" good in the production of the final output. Thus the production is:
Y (t)
=K
(t) a (h (t) RN (t)) t3 (J( (t) vI) l-a- t3 , (I)
o,/3 , Cl + (3
E (0,1),
where K (t) is the capital stock, RN (t) the intermediate "natural" good, I the labor and h (t) the technological knowledge or efficiency of RN (t) in the production of Y (t). From the total labor, the portion v E (0, 1) is devoted to the production of the final output. The remainder goes to the environmental R&D sector increasing the productivity RN according to the dynamic equation:
The model is stated as a differential game between North and South. Endogenous technological progress and the learning by doing process cause endogenous sustained growth for the economy. Thus, in contrast to the exogenous growth models that focus on the analysis of the steady state, the attention here is centered on the balanced path, i.e. the growth trajectory where all variables grow at constant rates. Since the economies are not typically located along the balanced paths, it is interesting to know how the model behaves in the transition towards the trajectories of sustainable growth, that is, economic policies leading to sustained growth in the North which does not deplete natural resource in the South. The transitional dynamics studies the dynamics of the model in the short and medium terms when this is not at the balanced path. The analysis of the transitional dynamics is important to show the losses and gains of either growth or welfare during the adaptation process as well as its duration, see (Hettich, 2000; Steger, 2000). The study can be analytically developed for the simplest models but more complex specifications require the implementation of numerical methods. The pioneer works of Mulligan and Sala-i-Martin (1991, 1993) develop a numerical algorithm to characterize the transitional dynamics of endogenous growth models.
h=1](l-v)hI,
(2)
77 > 0,
where I is constant. I The dynamics for the capital stock is given by the total production minus consumption and the cost of the intermediate "natural" good :
K = KO (hRN l (KvI)I-a- t3
- pRN -
C,
(3)
The North maximizes its intertemporal utility discounted at rate p > 0, fooo e- pt In c dt. It chooses the consumption, c, the demand for the intermediate "natural" good, RN, and the share of labor devoted to the final output sector, v, subject to (3) and (2), c,R N 2: 0 , v E (0,1). The South harvests from two different renewable resources. The extracted amounts make up a homogeneous input with which the South produces an intermediate "natural" good which is sold to the North to produce a final output. The dynamics of the natural resources is given by a logistic growth function and the hypothesis that the catch-per-unit effort is proportional to the stock level, see (Clark, 1990). Thus,
The plan of this paper is as follows: in Section 2 the Northern and Southern economies and their trade relationships are established. The focus is on the balanced path in Section 3 and in Section 4 the transitional dynamics is analyzed. Finally, conclusions are presented in Section 5.
SI = 7"ISI (1 - sI/ cC[) - qEIs I ,
(4)
'~2 = r2 82 (1- s2/cCII) - q (E - El) S2,
(5)
(6) Variables SI and S2 measure the resources' stocks. The intrinsic growth rates are given by the positive values rl and r2. The positive constants CCI and CCII are referred to as the carrying capacities. The positive constant q is the catchability coefficient, the same for both resources. El is the effort allocated to the first resource while E is the total effort for both resources.
2. THEMODEL The production function in the North depends on capital, labor and an intermediate "natural" good which is produced in the South and extracted from two renewable resources. The hypothesis of learning by doing means a positive effect of experience on the productivity of labor. That is, when a firm invests in physical capital its workers learn how to produce more efficiently. In addition to a sector that produces final output, following Bovenberg and Smulders (1995; 1996), an environmental R&D sector is introduced. This pure investment sector produces a technology that enhances
For the sake of tractability the production of the intermediate "natural" good depends not on the extraction effort of each resource, but on the total effort, E . It also depends on each resource's stock, Rs=(E,SI , S2)=Esfs~ ,
(7)
where () E (0,1) andE, <1>8" E8i > 0, i 1, 2. A higher stock of anyone of the resources leads to a
1
76
Henceforth time arguments are omitted.
greater production of the intermediate good but also a higher productivity of the total effort. The output elasticity, e, is identical for SI and S2.
with a constant v imply that h, k, c, p, mh and mK grow at the same rate, where mh and mK denote the shadow prices of the technological knowledge and of the capital stock respectively. As it is standard in this kind of models, new variables c = cl K, h = hi K, P = pi K and qh = mhlmK are defined. The balanced path of the original variables corresponds to the steady states of these new variables. However for the sake of tractability the number of variables is reduced assuming X = qhh. The equilibria for the dynamic equations written in terms of the new variables, c, X, P and R, correspond to a sustained growth path. However the dynamics for both the price and the quantity of the intermediate good depend on the evolution of the natural resources' stocks in the South, given by (4) and (5). It is easy to show that balanced paths exist. In fact two balanced paths appear with constant SI and S2. When S] reaches its carrying capacity, Sz must remain constant below its carrying capacity and vice versa. Furthermore, balanced paths with a fluctuating behavior of 81 and 82 satisfying equation (9) could exist. Next, the focus is on the behavior of SI and S2 for which the intermediate good remains constant.
The South has to decide the total harvesting effort, E and which part of this effort is used for resource I, El, and consequently for resource 2. It maximizes its stream of utility discounted at the rate p' > O. Utility is viewed as a logarithmic transformation of the monetary value of the total production in the South, In (pRs). From this utility and the expression of Rs in (7), the first order conditions will show that the optimal supply of the intermediate good is independent of the price, p. Thus the supply for this good is indeed equivalent to the amount sold to and used in the North, and pRs coincides with the total income in the South. Therefore, what the utility function states is the non-existence of an investment process in the South. All income is consumed. This region maximizes J~oo e-p't In (pRs) dt, choosing E and El subject to (4),(5) and (6), E 2: El 2: O. From the necessary conditions for optimality, the optimal paths for both regions and the equilibrium price for the traded good can be characterized. This price, p, stems from the equality between the South's supply and the North's demand.
A sustained growth path involves the use of a constant amount of intermediate good, whose price per unit of capital remains constant. However the two resources' stocks do not need to remain constant. Considering equation (9) looking at the dynamics of one of the two resources (let us say s]) the other resource's dynamics is also known. According to (6) and (9) the first resource's stock takes values between two bounds z :
3. THE BALANCED PATH
In an endogenous growth model the analysis focuses on the balanced path, defined as the growth trajectory where all variables grow at constant rates (which can be zero in some cases). The lab.or share devoted to the final output sector, v, takes values between zero and one, so it remains constant on the balanced path. The natural resources' stocks are positive and lower than their carrying capacities. Since the intermediate "natural" good production depends on the natural resources extracted which are limited, then in a balanced path, this good also remains constant. From now on this good is denoted by R, given that in equilibrium R = Rs = RN . The growth rate for this good is given by:
max {O, (cl> -
1.6.}
:s SI :s min {cl> 1.6., CcJ} .
The first resource will never be depleted if
eT] - p' -
r'2
> O.
() 0)
On the other hand, condition ()rz -
p'
>0
(11 )
guarantees that S2 will never be exhausted. In what follows, the optimal dynamical system for the South, as well as the dynamics for the first resource are analyzed. To make work easier, new variables are defined, PSi == m Si Si which can be understood as the total value of the i-th resource for the South. The dynamic system is then,
RIR=e{Tl (I-Sr/CCl)+T2(I-s2/cCII)}p'-T2S2/cCII .
CCll)
(8)
SI =
rlsl
(1- Sr/CCl) - qEIS],
A constant R leads to a linear and downward sloped relationship between natural resources,
PSI = PSI [p'
(9)
PS2 = PS2 [p'
+ Tl sI/ccIl - e , + r2 (cl> - .6.s 1 ) Icclll- (),
(12) ( 13)
(14)
where
e
where.6. > 0 and condition (Tl + T2) - p' > 0 implies a positive cl> and a non negative S2. If this inequality is not fulfilled (8) the intermediate good would fall indefinitely and no balanced path would be possible.
E El
The necessary conditions for optimality, the growth rates of the capital stock and of consumption, together
2
77
= 11 (qPS2) ' ={
0
E
if if
(15)
PSI - PS2 > 0, PSI - PS 2 < 0,
A similar expression can be written for
S2
(16)
PS 2 phase plane where E\ = E . The second column presents the initial conditions (LC) under which E\ switches in the way expressed in first column. These LC refer to points on, above or below the stable arm (s.a.) on the initial phase plane. From the analysis of the phase planes LC that do not allow the behavior described in first column are removed. Numerical simulations have been carried out for different parameter values and LC It has not been possible to remove those LC with a question mark for which there has been no success with numerical integration either. Tables I and 2 establish that after one or two changes of the optimal effort, El, from 0 to E or vice versa, this effort remains constant henceforth. When E1 = 0 takes place in the final step 81 converges to its carrying capacity while 82 tends to a positive value. Likewise, when the total effort is centered on the first resource in this last stage, 82 tends to its saturation level and 81 to a positive steady state.
and 82 has been replaced by its expression in (9). The optimal total effort is a decreasing function of the total value of the second resource's stock. The effort devoted to each resource is defined by a bang-bang function which assigns all the effort to the less valued resource. It can be shown that the dynamic system for 81, 82, PSI and PS 2 and the system (12)-(14) present identical steady states although the transitional dynamics are not necessarily the same.
Given that the optimal effort is a bang-bang function, the system is analyzed assuming a null extraction effort for the first natural resource on one hand, and a maximum effort for this resource on the other hand. This is a first step to understanding the complete evolution of the system. The remainder scenario is characterized by a shift from null (maximum) to maximum (null) effort either a finite or an infinite number of times. I. El = 0 To study the evolution of the system (12)-(14) the SI - Ps, and the 81 - PS 2 phase planes are analyzed separately. For each plane there is a saddle point equilibrium as well as a source. The saddle point is always attained at 81 = CCI. In the unstable equilibrium the first resource is completely depleted, 81 = O.
LC
0-+ E O-+E-+O
An indefinitely null effort El, consistent with the convergent paths towards the saddle point, is only possible for initial conditions placed either on or above the stable arm. In the first case, the paths converge to the saddle point where 81 indeed reaches cc,. In the second case, 8\ converges to its carrying capacity. Condition
p'
+ (1 + 8) Tl
-
(jT2
< 0,
LC
below s.a.""below s.a. (7) on s.a. "" on s.a. above s.a. "" on s.a. below s.a. "" below s.a.
LC
E-+O E-+O-+E
(17)
guarantees an uninterrupted null effort and a positive value of 82 at the stationary state. This condition implies T1 < T2 .
above s.a. above s.a. (7)
4. TRANSITIONAL DYNAMICS TO SUSTAINABILlTY
2. El = E Given the dynamics of the first resource and since E\ depends on Ps 2 , only the 81 - PS 2 phase plane can be studied. Condition (10) must be fulfilled, otherwise there would be a unique steady state where the first resource would be depleted. Thus two equilibria appear, a saddle point and a source.
In most endogenous growth models which take into account the environment, the analysis is restricted to the balanced path where all variables grow at their long-run rates, and the use of natural resources is sustainable. However, in the real world, there might be imbalances between different sectors, or the use of resources might be above their sustainable levels. In this case it is of interest to analyze the transition path to the sustainable growth solution.
For the saddle point to be reached, under condition ( 10), the stable arm is the only admissible path. In this case 8 2 reaches CCI I at the stationary state but 81 attains a value below CCI.
Deviations of variable X from its steady state value represent imbalances between the final output sector and the sector of innovation. Variable p measures imbalances in the price of the intermediate "natural" good. Variables R, 81 and 82 say if the exploitation of the natural resources is above or below their sustainable levels.
The next step is to study whether and under which initial conditions the corresponding stable arm is reached, when shifts in bang-bang function E\ from 0 to E and vice versa are allowed. Tables 1 and 2 focus on the cases where the first resource is not depleted. Table I (Table 2) gathers together these cases with an initial zero (total) effort on the first resource. Under condition (10) there are two steady states in the 8\ -
Two balanced paths have been characterized assuming a null effort for the first resource on the one hand and
78
a maximum effort for this resource on the other hand. The existence of transition paths converging on these two balanced paths is studied. For both scenarios, El = 0 and El = E, the steady state of the dynamic equations describing the motion of the relevant variables is a saddle point with a three-dimensional stable manifold. As a consequence, there are some imbalances in the relevant variables which can be corrected to catch up with the balanced path. Moreover, the path converging on the balanced path is not unique, but there is an infinite number of trajectories approaching it (all of which are over a three-dimensional stable manifold).
sI
Fig. I . Transition path for
Next, numerical simulations show which optimal policies direct the economies to the balanced path and how the growth rates of different variables behave along the transition periods. The time elimination method of Mulligan and Sala-i-Martin (1991,1993) is usually applied to a two-sector endogenous growth model, with only one negative eigenvalue. Here the algorithm is adapted to a model that presents three real negative eigenvalues. It must be implemented three times and the common behavior for all of them coincides with that of the path converging on the steady state. The values for the parameters are p = 0.08, p' = 0.1, a = (3 = 0.25, TJ = 0.2, Tl = 0.2, T2 = 0.4, CCl = 1,CCII = 2, q = 1, 8 = 0.5.
o
-0.5
-1
0.5
S]
Optimal policies must boost the R&D sector (1 - '/I has to be above its steady state value). They must also limit the use of the intermediate "natural" good which has to be below its long-run value, as Figure 2 shows. Along the transition R grows. The growth rates of physical capital and consumption in the North are lower than in the long-run and they grow along the transition. Conversely the growth ratc of technological knowledge is higher than in the longrun and it diminishes. The growth rate of the intermediate "natural" good is above its equilibrium level whereas the growth rate of its price is below the longrun value. The first one diminishes whereas the second one increases along the transition .
A long-run growth rate of 0.04 is obtained for the North. In the South, which does not accumulate capital, consumption grows at this same rate in the balanced path. The long-run values for the rest of variables are c* = 0.08, X* = 0.5, p* = 2.07, R* = 0.019, si = 1 and S2 = 0.3333, when El = O. For these values it * = 0.05. In the following figures, the horizontal axis represents the deviation of variable it from its steady state level it* . More precisely, In(itlit*) is represented in this axis, so the origin represents the balanced path. The vertical axis represents each relevant variable.
R
0.035
//
0.03 0 . 025 0.02
0.015 0.01
Assume an initial position where variable it, is below its steady state value (it < it*). There exists an imbalance between technological knowledge and physical capital in the North; the knowledge is scarcer. In the figures below the focus is on the trajectories approaching the origin from the left. Natural resources are also assumed to be below their long-run values. This could represent an industrialized Northern region with a high level of capital and natural resource intensive production technologies and a Southern region which overexploits its natural resources. Along an optimal path approaching the long-run solution, the ratio it = hi k grows as SI and S2 grow (see Figure I for SI)' That is, the Northern economy becomes more knowledge intensive while the resources' stocks increase in the South.
-1
-o.S
o
O.S
Fig. 2. Transition path for R On an optimal path, consumption in the South is proportional to final output in the North. The growth rate of the price increases with it, whereas the growth rate of R decreases, then the behavior of the growth rate of final output and consumption in the South will differ from some trajectories to others. The idea that more environmentally friendly production technologies will always diminish the final output growth rate can be rejected. Restrictions in the use of the intermediate "natural" good will not necessarily slow down the growth rate of consumption in the South. Reverse reasoning can be applied when it is initially above its steady state (it > it*) and natural resources are above their long-run values. Different elections
79
for the parameters give similar results. The outcomes when El = E are also similar.
REFERENCES Barro, R. and X. Sala-i-Martin (1995). Economic Growth. McGraw-Hill, New York. Bovenberg, A.L. and S.A. Smulders (1995). Environmental quality and pollution-augmenting technological change in a two-sectors endogenous growth model. Journal of Public Economics, 57, 369-391. Cabo, F; Escudero, E. and Martfn-Hemin, G. (1999). Ecological Technologies and Sustainable Development in a North-South Trade Model. In: Business & Economics for the 2 Jst Century (D. Kantarelis, Ed.). Vol. Ill, pp. 535-546. Business & Economics Society International. Chichilnisky, G. (1994). North-South trade and the Global Environment. American Economic Review, 84, 851-874. CIark, CW. (1990). Mathematical Bioeconomics. The Optimal Management of Renewable Resources. Wiley-Interscience Publications, New York . Copeland, B. and M.S. Taylor (1994). North-South trade and the environment. Quarterly Journal of Economics, 109, 755-787. Hettich, F (2000). Economic Growth and Environmental Policy. A Theoretical Approach. Edward Elgar, Cheltenham. Mulligan, CB . and X. Sala-i-Martin (1991). A note on the time-elimination method for solving recursive dynamic economic models. NBER Technical Working Paper, 116. Mulligan, CB. and X. Sala-i-Martin (1993). Transitional dynamics in two-sector models of endogenous growth . Quarterly Journal of Economics, 108, 739-773. Romer, P. (1986). Increasing returns and long-run growth. Journal of Political Economy, 94, 10021037. Steger, T. (2000). Transitional Dynamics alld Economic Growth in Developing Countries. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, Germany.
5. CONCLUSIONS A model of trade between an industrialized North and a developing South has been presented. Balanced paths exist that allow a sustained growth in the North and a permanent growth of consumption in the South without exhausting Southern natural resources. On a balanced path, physical capital, technological knowledge, consumption and the price of the intermediate "natural" good, grow at the same constant rate while the supply and demand of the intermediate good remain constant. However, the stocks of the two resources are not necessarily constant. For some balanced paths only one of the two resources is harvested, allowing the other to reach its carrying capacity. For other balanced paths the total exploitation effort interchanges from one resource to the other, although this will not happen continuously.
Transitional dynamics have been also studied. If there exists an imbalance between technological knowledge and physical capital in the North, the former being scarcer, and natural resources are below their longrun values, optimal policies must promote the R&D sector and limit the use of intermediate "natural" good in order to balance the ratio between technological knowledge and physical capital and to increase the stock of natural resources. Optimal policies will also affect the growth rates of the variables. Numerical simulations reject the idea that more environmentally friendly production technologies will always diminish the final output growth rate. Restrictions in the use of the intermediate "natural" good will not necessarily slow down the growth rate of consumption in the South.
Acknowledgments: The authors have been partially supported by project DGESIC PB98-0358.
80
ELSEVIER
IFAC PUBLICATIONS www.elsevier.com/ locate/ifac
SUSTAINABLE GROWTH IN A NORTH-SOUTH TRADE MODEL Francisco Cabo • , Guiomar Martin-Hernin •• , Maria Pilar Martinez-Garcia •••
• Dept. Econom(a Aplicada (Matemdticas). Avda. Valle Esgueva, 6, Valladolid 47011, Spain •• Dept. Econom(a Aplicada (Matemdticas). Avda. Valle Esgueva, 6, Valladolid 4701 1, Spain ••• Dept. Metodos Cuantitativos para la Econom(a. Campus del Espinardo, Murcia 30100, Spain
Abstract: A model with two different regions that trade with each other is presented. The South specializes in a resource intensive good while the North specializes in a capital intensive good. The North-South trade and the management of natural resources are modelled in a dynamic way so that sustainable economic growth can be analyzed . The existence of balanced paths that allow sustained growth in the North and permanent growth of consumption in the South without exhausting Southern natural resources is proved. The transitional dynamics to a balanced path is also studied. Copyright © 2001 IFAC Keywords : Differential games, economic systems, dynamic models, numerical simulation, steady states
only two different kinds. The stocks of these resources present different dynamic evolutions. The South harvests from these two resources and with the extracted amount it produces a resource intensive good. The North buys this good on the international market and uses it as an intermediate "natural" good to produce a capital intensive good.
I. INTRODUCTION
Most of the authors that study endogenous growth and the environment focus on an isolated country without taking into account trade relationships with other countries. Even those that consider several countries and settle a differential game do assume that all countries are identical, see (Hettich, 2000).
Within the endogenous growth literature, technological knowledge either augments labor productivity in the final output production function or expands product variety or raises product quality, see (Barro and Sala-i-Martin, 1995). On the contrary, in the presented paper technological knowledge enhances the productivity of an intermediate "natural" good. Following Bovenberg and Smulders (1995), an environmental R&D sector devoted to increasing the efficiency of the intermediate "natural" good in the North's output production is assumed.
In contrast to the previous works and following (Cabo et al., 1999) a model with two different regions that trade with each other is presented. Within this framework of North-South trade and environmental problems, following the static approach by Chichilnisky (1994) and Cope land and Taylor (1994), the South specializes in a resource intensive good. Nevertheless the North-South trade and the management of natural resources is modelled in a dynamic way so that sustainable economic growth can be analyzed. All possible renewable resources in the South are simplified to
75
Romer (1986) eliminates the diminishing returns to the factors, assuming that knowledge is a by-product of capital accumulation. A firm that increases its physical capital learns at the same time how to produce more efficiently. This is known as the learning by doing effect. Additionally, it is assumed that each firm's knowledge is a public good which spills over instantly across all other firms at no cost. This makes labor productivity dependent not on each firm's capital stock but on the whole economy's. These two combined effects are considered to specify how labor affects the North's production function.
efficiency in the use of the intermediate "natural" good in the production of the final output. Thus the production is:
Y (t)
=K
(t) a (h (t) RN (t)) t3 (J( (t) vI) l-a- t3 , (I)
o,/3 , Cl + (3
E (0,1),
where K (t) is the capital stock, RN (t) the intermediate "natural" good, I the labor and h (t) the technological knowledge or efficiency of RN (t) in the production of Y (t). From the total labor, the portion v E (0, 1) is devoted to the production of the final output. The remainder goes to the environmental R&D sector increasing the productivity RN according to the dynamic equation:
The model is stated as a differential game between North and South. Endogenous technological progress and the learning by doing process cause endogenous sustained growth for the economy. Thus, in contrast to the exogenous growth models that focus on the analysis of the steady state, the attention here is centered on the balanced path, i.e. the growth trajectory where all variables grow at constant rates. Since the economies are not typically located along the balanced paths, it is interesting to know how the model behaves in the transition towards the trajectories of sustainable growth, that is, economic policies leading to sustained growth in the North which does not deplete natural resource in the South. The transitional dynamics studies the dynamics of the model in the short and medium terms when this is not at the balanced path. The analysis of the transitional dynamics is important to show the losses and gains of either growth or welfare during the adaptation process as well as its duration, see (Hettich, 2000; Steger, 2000). The study can be analytically developed for the simplest models but more complex specifications require the implementation of numerical methods. The pioneer works of Mulligan and Sala-i-Martin (1991, 1993) develop a numerical algorithm to characterize the transitional dynamics of endogenous growth models.
h=1](l-v)hI,
(2)
77 > 0,
where I is constant. I The dynamics for the capital stock is given by the total production minus consumption and the cost of the intermediate "natural" good :
K = KO (hRN l (KvI)I-a- t3
- pRN -
C,
(3)
The North maximizes its intertemporal utility discounted at rate p > 0, fooo e- pt In c dt. It chooses the consumption, c, the demand for the intermediate "natural" good, RN, and the share of labor devoted to the final output sector, v, subject to (3) and (2), c,R N 2: 0 , v E (0,1). The South harvests from two different renewable resources. The extracted amounts make up a homogeneous input with which the South produces an intermediate "natural" good which is sold to the North to produce a final output. The dynamics of the natural resources is given by a logistic growth function and the hypothesis that the catch-per-unit effort is proportional to the stock level, see (Clark, 1990). Thus,
The plan of this paper is as follows: in Section 2 the Northern and Southern economies and their trade relationships are established. The focus is on the balanced path in Section 3 and in Section 4 the transitional dynamics is analyzed. Finally, conclusions are presented in Section 5.
SI = 7"ISI (1 - sI/ cC[) - qEIs I ,
(4)
'~2 = r2 82 (1- s2/cCII) - q (E - El) S2,
(5)
(6) Variables SI and S2 measure the resources' stocks. The intrinsic growth rates are given by the positive values rl and r2. The positive constants CCI and CCII are referred to as the carrying capacities. The positive constant q is the catchability coefficient, the same for both resources. El is the effort allocated to the first resource while E is the total effort for both resources.
2. THEMODEL The production function in the North depends on capital, labor and an intermediate "natural" good which is produced in the South and extracted from two renewable resources. The hypothesis of learning by doing means a positive effect of experience on the productivity of labor. That is, when a firm invests in physical capital its workers learn how to produce more efficiently. In addition to a sector that produces final output, following Bovenberg and Smulders (1995; 1996), an environmental R&D sector is introduced. This pure investment sector produces a technology that enhances
For the sake of tractability the production of the intermediate "natural" good depends not on the extraction effort of each resource, but on the total effort, E . It also depends on each resource's stock, Rs=(E,SI , S2)=Esfs~ ,
(7)
where () E (0,1) and
1
76
Henceforth time arguments are omitted.
greater production of the intermediate good but also a higher productivity of the total effort. The output elasticity, e, is identical for SI and S2.
with a constant v imply that h, k, c, p, mh and mK grow at the same rate, where mh and mK denote the shadow prices of the technological knowledge and of the capital stock respectively. As it is standard in this kind of models, new variables c = cl K, h = hi K, P = pi K and qh = mhlmK are defined. The balanced path of the original variables corresponds to the steady states of these new variables. However for the sake of tractability the number of variables is reduced assuming X = qhh. The equilibria for the dynamic equations written in terms of the new variables, c, X, P and R, correspond to a sustained growth path. However the dynamics for both the price and the quantity of the intermediate good depend on the evolution of the natural resources' stocks in the South, given by (4) and (5). It is easy to show that balanced paths exist. In fact two balanced paths appear with constant SI and S2. When S] reaches its carrying capacity, Sz must remain constant below its carrying capacity and vice versa. Furthermore, balanced paths with a fluctuating behavior of 81 and 82 satisfying equation (9) could exist. Next, the focus is on the behavior of SI and S2 for which the intermediate good remains constant.
The South has to decide the total harvesting effort, E and which part of this effort is used for resource I, El, and consequently for resource 2. It maximizes its stream of utility discounted at the rate p' > O. Utility is viewed as a logarithmic transformation of the monetary value of the total production in the South, In (pRs). From this utility and the expression of Rs in (7), the first order conditions will show that the optimal supply of the intermediate good is independent of the price, p. Thus the supply for this good is indeed equivalent to the amount sold to and used in the North, and pRs coincides with the total income in the South. Therefore, what the utility function states is the non-existence of an investment process in the South. All income is consumed. This region maximizes J~oo e-p't In (pRs) dt, choosing E and El subject to (4),(5) and (6), E 2: El 2: O. From the necessary conditions for optimality, the optimal paths for both regions and the equilibrium price for the traded good can be characterized. This price, p, stems from the equality between the South's supply and the North's demand.
A sustained growth path involves the use of a constant amount of intermediate good, whose price per unit of capital remains constant. However the two resources' stocks do not need to remain constant. Considering equation (9) looking at the dynamics of one of the two resources (let us say s]) the other resource's dynamics is also known. According to (6) and (9) the first resource's stock takes values between two bounds z :
3. THE BALANCED PATH
In an endogenous growth model the analysis focuses on the balanced path, defined as the growth trajectory where all variables grow at constant rates (which can be zero in some cases). The lab.or share devoted to the final output sector, v, takes values between zero and one, so it remains constant on the balanced path. The natural resources' stocks are positive and lower than their carrying capacities. Since the intermediate "natural" good production depends on the natural resources extracted which are limited, then in a balanced path, this good also remains constant. From now on this good is denoted by R, given that in equilibrium R = Rs = RN . The growth rate for this good is given by:
max {O, (cl> -
1.6.}
:s SI :s min {cl> 1.6., CcJ} .
The first resource will never be depleted if
eT] - p' -
r'2
> O.
() 0)
On the other hand, condition ()rz -
p'
>0
(11 )
guarantees that S2 will never be exhausted. In what follows, the optimal dynamical system for the South, as well as the dynamics for the first resource are analyzed. To make work easier, new variables are defined, PSi == m Si Si which can be understood as the total value of the i-th resource for the South. The dynamic system is then,
RIR=e{Tl (I-Sr/CCl)+T2(I-s2/cCII)}p'-T2S2/cCII .
CCll)
(8)
SI =
rlsl
(1- Sr/CCl) - qEIS],
A constant R leads to a linear and downward sloped relationship between natural resources,
PSI = PSI [p'
(9)
PS2 = PS2 [p'
+ Tl sI/ccIl - e , + r2 (cl> - .6.s 1 ) Icclll- (),
(12) ( 13)
(14)
where
e
where.6. > 0 and condition (Tl + T2) - p' > 0 implies a positive cl> and a non negative S2. If this inequality is not fulfilled (8) the intermediate good would fall indefinitely and no balanced path would be possible.
E El
The necessary conditions for optimality, the growth rates of the capital stock and of consumption, together
2
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= 11 (qPS2) ' ={
0
E
if if
(15)
PSI - PS2 > 0, PSI - PS 2 < 0,
A similar expression can be written for
S2
(16)
PS 2 phase plane where E\ = E . The second column presents the initial conditions (LC) under which E\ switches in the way expressed in first column. These LC refer to points on, above or below the stable arm (s.a.) on the initial phase plane. From the analysis of the phase planes LC that do not allow the behavior described in first column are removed. Numerical simulations have been carried out for different parameter values and LC It has not been possible to remove those LC with a question mark for which there has been no success with numerical integration either. Tables I and 2 establish that after one or two changes of the optimal effort, El, from 0 to E or vice versa, this effort remains constant henceforth. When E1 = 0 takes place in the final step 81 converges to its carrying capacity while 82 tends to a positive value. Likewise, when the total effort is centered on the first resource in this last stage, 82 tends to its saturation level and 81 to a positive steady state.
and 82 has been replaced by its expression in (9). The optimal total effort is a decreasing function of the total value of the second resource's stock. The effort devoted to each resource is defined by a bang-bang function which assigns all the effort to the less valued resource. It can be shown that the dynamic system for 81, 82, PSI and PS 2 and the system (12)-(14) present identical steady states although the transitional dynamics are not necessarily the same.
Given that the optimal effort is a bang-bang function, the system is analyzed assuming a null extraction effort for the first natural resource on one hand, and a maximum effort for this resource on the other hand. This is a first step to understanding the complete evolution of the system. The remainder scenario is characterized by a shift from null (maximum) to maximum (null) effort either a finite or an infinite number of times. I. El = 0 To study the evolution of the system (12)-(14) the SI - Ps, and the 81 - PS 2 phase planes are analyzed separately. For each plane there is a saddle point equilibrium as well as a source. The saddle point is always attained at 81 = CCI. In the unstable equilibrium the first resource is completely depleted, 81 = O.
LC
0-+ E O-+E-+O
An indefinitely null effort El, consistent with the convergent paths towards the saddle point, is only possible for initial conditions placed either on or above the stable arm. In the first case, the paths converge to the saddle point where 81 indeed reaches cc,. In the second case, 8\ converges to its carrying capacity. Condition
p'
+ (1 + 8) Tl
-
(jT2
< 0,
LC
below s.a.""below s.a. (7) on s.a. "" on s.a. above s.a. "" on s.a. below s.a. "" below s.a.
LC
E-+O E-+O-+E
(17)
guarantees an uninterrupted null effort and a positive value of 82 at the stationary state. This condition implies T1 < T2 .
above s.a. above s.a. (7)
4. TRANSITIONAL DYNAMICS TO SUSTAINABILlTY
2. El = E Given the dynamics of the first resource and since E\ depends on Ps 2 , only the 81 - PS 2 phase plane can be studied. Condition (10) must be fulfilled, otherwise there would be a unique steady state where the first resource would be depleted. Thus two equilibria appear, a saddle point and a source.
In most endogenous growth models which take into account the environment, the analysis is restricted to the balanced path where all variables grow at their long-run rates, and the use of natural resources is sustainable. However, in the real world, there might be imbalances between different sectors, or the use of resources might be above their sustainable levels. In this case it is of interest to analyze the transition path to the sustainable growth solution.
For the saddle point to be reached, under condition ( 10), the stable arm is the only admissible path. In this case 8 2 reaches CCI I at the stationary state but 81 attains a value below CCI.
Deviations of variable X from its steady state value represent imbalances between the final output sector and the sector of innovation. Variable p measures imbalances in the price of the intermediate "natural" good. Variables R, 81 and 82 say if the exploitation of the natural resources is above or below their sustainable levels.
The next step is to study whether and under which initial conditions the corresponding stable arm is reached, when shifts in bang-bang function E\ from 0 to E and vice versa are allowed. Tables 1 and 2 focus on the cases where the first resource is not depleted. Table I (Table 2) gathers together these cases with an initial zero (total) effort on the first resource. Under condition (10) there are two steady states in the 8\ -
Two balanced paths have been characterized assuming a null effort for the first resource on the one hand and
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a maximum effort for this resource on the other hand. The existence of transition paths converging on these two balanced paths is studied. For both scenarios, El = 0 and El = E, the steady state of the dynamic equations describing the motion of the relevant variables is a saddle point with a three-dimensional stable manifold. As a consequence, there are some imbalances in the relevant variables which can be corrected to catch up with the balanced path. Moreover, the path converging on the balanced path is not unique, but there is an infinite number of trajectories approaching it (all of which are over a three-dimensional stable manifold).
sI
Fig. I . Transition path for
Next, numerical simulations show which optimal policies direct the economies to the balanced path and how the growth rates of different variables behave along the transition periods. The time elimination method of Mulligan and Sala-i-Martin (1991,1993) is usually applied to a two-sector endogenous growth model, with only one negative eigenvalue. Here the algorithm is adapted to a model that presents three real negative eigenvalues. It must be implemented three times and the common behavior for all of them coincides with that of the path converging on the steady state. The values for the parameters are p = 0.08, p' = 0.1, a = (3 = 0.25, TJ = 0.2, Tl = 0.2, T2 = 0.4, CCl = 1,CCII = 2, q = 1, 8 = 0.5.
o
-0.5
-1
0.5
S]
Optimal policies must boost the R&D sector (1 - '/I has to be above its steady state value). They must also limit the use of the intermediate "natural" good which has to be below its long-run value, as Figure 2 shows. Along the transition R grows. The growth rates of physical capital and consumption in the North are lower than in the long-run and they grow along the transition. Conversely the growth ratc of technological knowledge is higher than in the longrun and it diminishes. The growth rate of the intermediate "natural" good is above its equilibrium level whereas the growth rate of its price is below the longrun value. The first one diminishes whereas the second one increases along the transition .
A long-run growth rate of 0.04 is obtained for the North. In the South, which does not accumulate capital, consumption grows at this same rate in the balanced path. The long-run values for the rest of variables are c* = 0.08, X* = 0.5, p* = 2.07, R* = 0.019, si = 1 and S2 = 0.3333, when El = O. For these values it * = 0.05. In the following figures, the horizontal axis represents the deviation of variable it from its steady state level it* . More precisely, In(itlit*) is represented in this axis, so the origin represents the balanced path. The vertical axis represents each relevant variable.
R
0.035
//
0.03 0 . 025 0.02
0.015 0.01
Assume an initial position where variable it, is below its steady state value (it < it*). There exists an imbalance between technological knowledge and physical capital in the North; the knowledge is scarcer. In the figures below the focus is on the trajectories approaching the origin from the left. Natural resources are also assumed to be below their long-run values. This could represent an industrialized Northern region with a high level of capital and natural resource intensive production technologies and a Southern region which overexploits its natural resources. Along an optimal path approaching the long-run solution, the ratio it = hi k grows as SI and S2 grow (see Figure I for SI)' That is, the Northern economy becomes more knowledge intensive while the resources' stocks increase in the South.
-1
-o.S
o
O.S
Fig. 2. Transition path for R On an optimal path, consumption in the South is proportional to final output in the North. The growth rate of the price increases with it, whereas the growth rate of R decreases, then the behavior of the growth rate of final output and consumption in the South will differ from some trajectories to others. The idea that more environmentally friendly production technologies will always diminish the final output growth rate can be rejected. Restrictions in the use of the intermediate "natural" good will not necessarily slow down the growth rate of consumption in the South. Reverse reasoning can be applied when it is initially above its steady state (it > it*) and natural resources are above their long-run values. Different elections
79
for the parameters give similar results. The outcomes when El = E are also similar.
REFERENCES Barro, R. and X. Sala-i-Martin (1995). Economic Growth. McGraw-Hill, New York. Bovenberg, A.L. and S.A. Smulders (1995). Environmental quality and pollution-augmenting technological change in a two-sectors endogenous growth model. Journal of Public Economics, 57, 369-391. Cabo, F; Escudero, E. and Martfn-Hemin, G. (1999). Ecological Technologies and Sustainable Development in a North-South Trade Model. In: Business & Economics for the 2 Jst Century (D. Kantarelis, Ed.). Vol. Ill, pp. 535-546. Business & Economics Society International. Chichilnisky, G. (1994). North-South trade and the Global Environment. American Economic Review, 84, 851-874. CIark, CW. (1990). Mathematical Bioeconomics. The Optimal Management of Renewable Resources. Wiley-Interscience Publications, New York . Copeland, B. and M.S. Taylor (1994). North-South trade and the environment. Quarterly Journal of Economics, 109, 755-787. Hettich, F (2000). Economic Growth and Environmental Policy. A Theoretical Approach. Edward Elgar, Cheltenham. Mulligan, CB . and X. Sala-i-Martin (1991). A note on the time-elimination method for solving recursive dynamic economic models. NBER Technical Working Paper, 116. Mulligan, CB. and X. Sala-i-Martin (1993). Transitional dynamics in two-sector models of endogenous growth . Quarterly Journal of Economics, 108, 739-773. Romer, P. (1986). Increasing returns and long-run growth. Journal of Political Economy, 94, 10021037. Steger, T. (2000). Transitional Dynamics alld Economic Growth in Developing Countries. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, Germany.
5. CONCLUSIONS A model of trade between an industrialized North and a developing South has been presented. Balanced paths exist that allow a sustained growth in the North and a permanent growth of consumption in the South without exhausting Southern natural resources. On a balanced path, physical capital, technological knowledge, consumption and the price of the intermediate "natural" good, grow at the same constant rate while the supply and demand of the intermediate good remain constant. However, the stocks of the two resources are not necessarily constant. For some balanced paths only one of the two resources is harvested, allowing the other to reach its carrying capacity. For other balanced paths the total exploitation effort interchanges from one resource to the other, although this will not happen continuously.
Transitional dynamics have been also studied. If there exists an imbalance between technological knowledge and physical capital in the North, the former being scarcer, and natural resources are below their longrun values, optimal policies must promote the R&D sector and limit the use of intermediate "natural" good in order to balance the ratio between technological knowledge and physical capital and to increase the stock of natural resources. Optimal policies will also affect the growth rates of the variables. Numerical simulations reject the idea that more environmentally friendly production technologies will always diminish the final output growth rate. Restrictions in the use of the intermediate "natural" good will not necessarily slow down the growth rate of consumption in the South.
Acknowledgments: The authors have been partially supported by project DGESIC PB98-0358.
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