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Resource extraction, international trade and production with the resource
Resource extraction, international trade and production with the resource A. Aydin Cecen
The paper analyses the optimal growth of a small economy that extracts an exhaustible resource both for domestic production and export purposes. It is shown that when the exogenously given resource price is constant, extraction for export will stop in finite time whereas extractionfor domestic production will continue. Yet, if the resource price increases exponentially, extraction will continue to support both sectors. Keywords: Author to supply
One of the distinguishing features of exhaustible resources is the uneven distribution of their stocks among nations. Due to this fact, international trade plays a crucial role in the transfer of these resources from resource-rich to resource-poor economies. Despite the voluminous literature on exhaustible resources, the number of studies that investigate the dynamic properties of resource extraction in the framework of international trade is rather modest. Vousden [7], Long [S] and Kemp and Suzuki [4] among others, examined, under different specifications, the long-run behaviour of resource economies engaged in international trade. Yet, for the most part, these models omit capital accumulation, which makes them inadequate in terms of scrutinizing optimal growth possibilities open to those economies that are dependent on natural resources in building their domestic capital stocks. More comprehensive and realistic studies came from Aarrested [ 1] and Dasgupta, Eastwood and Heal [3] that account both for capital accumulation and resource depletion as well as technical change and foreign borrowing. More recently, Moussavian [ 61 analysed the optimal extraction policies in the presence of non-traded goods. The author is Assistant Professor, University, Department of Economics, 48859, USA.
Central Michigan Mt Pleasant, MI
I amgrateful to R. A. Becker, P. Kuznets and W. Winston of Indiana University for their valuable comments. I also benefited detailed criticisms of Professor A. C. Chiang. Final manuscript
254
received 21 March
1991.
from the
The purpose of this paper is to fill the gap in the literature by developing a more realistic model that can address the capital accumulation and resource extraction policies when the resource is simultaneously used as an input and export product.
The model A simple one sector growth model is used to describe the optimal behaviour of a small economy that extracts the resource both for export and domestic production purposes. In other words, the economy is assumed to utilize part of the resource extracted as an input, in addition to capital and labour, and exports the rest of it in exchange for foreign consumption-investment products. Thus, the country faces the problem of allocating the resource optimally between domestic production and export. It is intuitively obvious that this optimal program would depend on the international price of the resource (or marginal export revenue), marginal product of the resource in domestic production and the substitution possibilities between capital and the resource input. On the other hand, the input properties of the resource are of crucial importance here since the length of the extraction period is determined by the specifications of the aggregate production function of the economy. In light of the above discussion, the model can be formalized in the following way. The aggregate output of the country is represented by a neoclassical
0140/9883/91/040254-04
0
1991 Butterworth-Heinemann
Ltd
Resource
production
function
extraction,
international
trade
and production
The optimality are:
x:
x = F(K,, L,, R,) with F, > 0, F, > 0, F, > 0,
with the resource:
conditions
and the costate equations
U’(C,) = E., F,,
< 0, F,,
< 0, F,,
< 0
(5)
(1) i_rf2 = jV2if R, > 0
where R, is the flow of resource input, K, and L, are, as usual, capital and labour. On the other hand, it is also assumed that the resource input is essential, ie’
S, > 0, the
’ [R, + R;]ds s
if R; = 0
(7b)
r, + p,R; = C, + I(-, + SK,
(3)
Equation (3) expresses the fact that domestic production plus the proceeds from resource export should be sufficient to cover total domestic consumption C, and domestic gross investment (K, + SK) where 6 is the constant exponential rate of depreciation. This form of the budget constraint implies that the current account is always balanced and resource extraction is costless. Given a constant social discount rate p, the optimization problem of the economy is to maximize the present value of total social welfare subject to the dynamic constraints (2) and (3) over an infinite horizon. Here the social welfare is represented by a utility function U(C) whose argument is per capita consumption and it has the following well known properties: U’ > 0, CJ” < 0, U’(0) = co, U’( co) = 0 Finally, without loss of generality, labour force is constant.* The current-value Hamiltonian
we assume that the of the system is:
R,) + p,R; - C, -
- %,[R, + R;]
dK,l
2, = 2z(O)ep’
(8b)
Finally,
and
f2
af
ZZ
8R’
the transversality
lim edpfill 1+lx
conditions
2 0; lim e-P’i.,,K, f-)rn
= 0
lim e-P’%21 2 0; lim eKPfA2J1 = 0 r+oo l+cc
are: (9a)
(9b)
Equation (5) is the standard result that says that along the optimal path the marginal utility of per capita consumption is equal to the shadow price of capital per worker. On the other hand, as long as R, > 0 and RI > 0, using Equations (6a) and (7a), we can write, .f* =
Pt
(10)
Equation (10) requires that along the optimal path, the marginal product of the resource be equal to the relative market price of the resource: this is an obvious condition for static efficiency which expresses the equality of marginal export revenue and the marginal contribution of the resource to domestic output. Whether this condition can be sustained indefinitely depends on the behaviour of the marginal product of the resource .r; and the exogenously given relative resource price pt as t + co. We shall tackle this question under different specifications of pt. On the other hand, differentiating Equation (6a) logarithmically with respect to time and using Equations (8a) and (8b) we obtain (11)
variables.
‘Dasgupta and Heal [2] discuss the exact meaning of an essential input. They define an exhaustible resource as inessential if there is a feasible program [R,, R,, C,],“= ,, along which consumption is bounded away from zero. ‘An exponential labour growth can be introduced without changing the results.
ECONOMICS
(8a)
(4)
where ;1r and A, are the costate
ENERGY
/PI = 1.[p + 6 -f;]
where fi = g
where R; is the flow of resource export. Furthermore, the small country assumption implies that the relative market price of the resource pt is exogenously given. Hence the intertemporal budget constraint can be written as:
(6b) (7a)
(2)
0
H = U(C,) + 2, Cf(K,
ij”R,=O
< 2,
Given the initial stock of the resource remaining resource stock S, is simply,
(6a)
iIpt = iz ifRi > 0
F(K,, L,, 0) = 0
S, = S, -
A. A. Cecen
October
1991
Equation (11) is condition, which arbitrage condition an additional stock using an additional two assets in this exhaustible resource,
the well-known Solow-Stiglitz is essentially an intertemporal between the returns from holding of the resource and those from amount of capital. Since there are open economy, capital and the this dynamic efficiency condition
255
Resource
extraction, international trade and production with the resource: A. A. Cecen
requires that the returns to these two assets be equal along the optimal path, net of depreciation. Given these necessary conditions, we can proceed to analyse the optimal behaviour ofthe economy under different specifications of the relative market price of the resource. Constant
resource price: pt = p
If the relative price of the resource is constant over time, ie pr = p, the possibility of the opening of trade depends on the value of &, p and 3LZtat t = 0. If: &(O)j
= I,(O)
or, from Equation fi(0)
(12) (10)
= P
then a portion of the extracted resource will be exported in exchange for other foreign investmentconsumption goods for a certain period of time. Since Equation ( 12) compares simply the exogenously given constant price with the marginal product of the resource ,f2 at time t = 0, it follows that when the marginal export revenue, @, is less than the marginal product of the resource in domestic production, ie p <.f*(O), the economy will not extract its resource for export purposes. This situation is basically the violation of the static efficiency condition expressed by Equation (IO), which shows that Ri = 0 when E.r(O)p < j.,(O). Consequently the resource economy will behave, from the very beginning, like a closed economy scrutinized by Dasgupta and Heal [2]. Moreover, considering Equation (7a) together with Equation (8b), we can write, ni,p = L2(0)eP’ if R; > 0
(13)
As t-+cC, the shadow price of the resource, A,, will go to infinity as shown by Equation (8b) in Equation ( 13). Yet, the left-hand side of Equation ( 13) does not tend to infinity as t-+cm because from Equation (5) we have U( C,) = A1 and the assumption of U’( C,) -+ co as C, -+ 0 guarantees that C, > 0 along the optimal path. Hence at a certain point in time, say t = t* the economy will switch to Equation (7b), ie Ai,p < & = for i.,(O)e”’ with RI = 0. Thus at t = t*, extraction export will stop. Therefore: Proposition I. If the exogenously given relative resource price is constant over time and if trade opens, resource export cannot last forever: there will be finite time t* < co when the economy will stop extracting the resource for export and will continue to extract the resource solely for domestic production purposes.
[2]. As is demonstrated in their paper, the essentiality of the resource for production (hence its exhaustion in finite time or infinity) is determined by the magnitude of the elasticity of substitution between capital and the resource, as well as the behaviour of the marginal product of the resource input. If positive output is possible in the absence of the resource input (ie F( K,, L,, 0) > 0) and if the marginal product of the resource is bounded from above as R,-+m 0, then it will be optimal for the economy to deplete the remaining stock of the resource in finite time and continue to produce only with the two traditional inputs, labour and capital. On the other hand, if either of these conditions fails, which is our case since we assumed that F( K,, L,, 0) = 0, then asymptotic depletion is optimal. Exponentially
When the exogenously given resource price increases exponentially at a constant rate & following the same logic above, it is easy to see that the interior solution Equation (7a) can be sustained forever since as t + co, from Equations (7a) and (8b) together with pt = p,,e@, Equation (13) will hold. Hence: Proposition II. If the relative resource price is exponentially increasing at a constant rate, resource extraction both for domestic production and export will continue forever as the resource is asymptotically exhausted.
Summary and conclusion This paper strived to analyse the optimal behaviour of a small resource economy that extracts its exhaustible resource both for domestic production and export purposes. We demonstrated that when the exogenously given relative resource price is constant over time, if trade opens, resource extraction for export would come to an end in finite time. After the economy stops exporting the resource, its optimal growth will be determined simultaneously by the elasticity of substitution between capital and the resource input and the dynamic behaviour of the marginal product of the resource, as depicted by Dasgupta and Heal 123. On the other hand, if the resource price has an exponential trend, we showed that resource extraction would continue forever to support domestic production and the export sector, as the resource is asymptotically depleted.
References 1
For t > t*, the optimal behaviour of the economy will be similar to the one analysed by Dasgupta and Heal
256
increasing resource price: pt = poePf
J. Aarrestad, ‘Optimal savings and exhaustible resource in an open economy’, Journal of Economic Theory, 1978, pp 163-179.
extraction
ENERGY
ECONOMICS
October
199 1
Resource extraction, 2
3
4
P. Dasgupta and G. Heal, ‘Optimal depletion of exhaustible resources’, Review L$ Economic Studies, Symposium, 1974, pp 3-28. P. Dasgupta, R. Eastwood and G. Heal, ‘Resource management in a trading economy’, Quarterly Journal of Economics, 1978, pp 297-306. M. C. Kemp and H. Suzuki, ‘International trade with a wasting but possibly replenishible resource’, International Economic Review, 1975, pp 712-732.
ENERGY
ECONOMICS
October
199 1
international 5
6
7
trade and production
with the resource:
A. A. Cecen
N. V. Long, ‘International borrowing for resource extraction’, International Economic Review, 1974. pp 168-183. M. Moussavian, ‘Growth rates with an exhaustible resource and home goods’, Journal of International Economics, 1985, pp 281-299. N. Vousden, ‘International trade and exhaustible resources: a theoretical model’, International Economic Ret’iew, 1976, pp 149-167.
257
The paper analyses the optimal growth of a small economy that extracts an exhaustible resource both for domestic production and export purposes. It is shown that when the exogenously given resource price is constant, extraction for export will stop in finite time whereas extractionfor domestic production will continue. Yet, if the resource price increases exponentially, extraction will continue to support both sectors. Keywords: Author to supply
One of the distinguishing features of exhaustible resources is the uneven distribution of their stocks among nations. Due to this fact, international trade plays a crucial role in the transfer of these resources from resource-rich to resource-poor economies. Despite the voluminous literature on exhaustible resources, the number of studies that investigate the dynamic properties of resource extraction in the framework of international trade is rather modest. Vousden [7], Long [S] and Kemp and Suzuki [4] among others, examined, under different specifications, the long-run behaviour of resource economies engaged in international trade. Yet, for the most part, these models omit capital accumulation, which makes them inadequate in terms of scrutinizing optimal growth possibilities open to those economies that are dependent on natural resources in building their domestic capital stocks. More comprehensive and realistic studies came from Aarrested [ 1] and Dasgupta, Eastwood and Heal [3] that account both for capital accumulation and resource depletion as well as technical change and foreign borrowing. More recently, Moussavian [ 61 analysed the optimal extraction policies in the presence of non-traded goods. The author is Assistant Professor, University, Department of Economics, 48859, USA.
Central Michigan Mt Pleasant, MI
I amgrateful to R. A. Becker, P. Kuznets and W. Winston of Indiana University for their valuable comments. I also benefited detailed criticisms of Professor A. C. Chiang. Final manuscript
254
received 21 March
1991.
from the
The purpose of this paper is to fill the gap in the literature by developing a more realistic model that can address the capital accumulation and resource extraction policies when the resource is simultaneously used as an input and export product.
The model A simple one sector growth model is used to describe the optimal behaviour of a small economy that extracts the resource both for export and domestic production purposes. In other words, the economy is assumed to utilize part of the resource extracted as an input, in addition to capital and labour, and exports the rest of it in exchange for foreign consumption-investment products. Thus, the country faces the problem of allocating the resource optimally between domestic production and export. It is intuitively obvious that this optimal program would depend on the international price of the resource (or marginal export revenue), marginal product of the resource in domestic production and the substitution possibilities between capital and the resource input. On the other hand, the input properties of the resource are of crucial importance here since the length of the extraction period is determined by the specifications of the aggregate production function of the economy. In light of the above discussion, the model can be formalized in the following way. The aggregate output of the country is represented by a neoclassical
0140/9883/91/040254-04
0
1991 Butterworth-Heinemann
Ltd
Resource
production
function
extraction,
international
trade
and production
The optimality are:
x:
x = F(K,, L,, R,) with F, > 0, F, > 0, F, > 0,
with the resource:
conditions
and the costate equations
U’(C,) = E., F,,
< 0, F,,
< 0, F,,
< 0
(5)
(1) i_rf2 = jV2if R, > 0
where R, is the flow of resource input, K, and L, are, as usual, capital and labour. On the other hand, it is also assumed that the resource input is essential, ie’
S, > 0, the
’ [R, + R;]ds s
if R; = 0
(7b)
r, + p,R; = C, + I(-, + SK,
(3)
Equation (3) expresses the fact that domestic production plus the proceeds from resource export should be sufficient to cover total domestic consumption C, and domestic gross investment (K, + SK) where 6 is the constant exponential rate of depreciation. This form of the budget constraint implies that the current account is always balanced and resource extraction is costless. Given a constant social discount rate p, the optimization problem of the economy is to maximize the present value of total social welfare subject to the dynamic constraints (2) and (3) over an infinite horizon. Here the social welfare is represented by a utility function U(C) whose argument is per capita consumption and it has the following well known properties: U’ > 0, CJ” < 0, U’(0) = co, U’( co) = 0 Finally, without loss of generality, labour force is constant.* The current-value Hamiltonian
we assume that the of the system is:
R,) + p,R; - C, -
- %,[R, + R;]
dK,l
2, = 2z(O)ep’
(8b)
Finally,
and
f2
af
ZZ
8R’
the transversality
lim edpfill 1+lx
conditions
2 0; lim e-P’i.,,K, f-)rn
= 0
lim e-P’%21 2 0; lim eKPfA2J1 = 0 r+oo l+cc
are: (9a)
(9b)
Equation (5) is the standard result that says that along the optimal path the marginal utility of per capita consumption is equal to the shadow price of capital per worker. On the other hand, as long as R, > 0 and RI > 0, using Equations (6a) and (7a), we can write, .f* =
Pt
(10)
Equation (10) requires that along the optimal path, the marginal product of the resource be equal to the relative market price of the resource: this is an obvious condition for static efficiency which expresses the equality of marginal export revenue and the marginal contribution of the resource to domestic output. Whether this condition can be sustained indefinitely depends on the behaviour of the marginal product of the resource .r; and the exogenously given relative resource price pt as t + co. We shall tackle this question under different specifications of pt. On the other hand, differentiating Equation (6a) logarithmically with respect to time and using Equations (8a) and (8b) we obtain (11)
variables.
‘Dasgupta and Heal [2] discuss the exact meaning of an essential input. They define an exhaustible resource as inessential if there is a feasible program [R,, R,, C,],“= ,, along which consumption is bounded away from zero. ‘An exponential labour growth can be introduced without changing the results.
ECONOMICS
(8a)
(4)
where ;1r and A, are the costate
ENERGY
/PI = 1.[p + 6 -f;]
where fi = g
where R; is the flow of resource export. Furthermore, the small country assumption implies that the relative market price of the resource pt is exogenously given. Hence the intertemporal budget constraint can be written as:
(6b) (7a)
(2)
0
H = U(C,) + 2, Cf(K,
ij”R,=O
< 2,
Given the initial stock of the resource remaining resource stock S, is simply,
(6a)
iIpt = iz ifRi > 0
F(K,, L,, 0) = 0
S, = S, -
A. A. Cecen
October
1991
Equation (11) is condition, which arbitrage condition an additional stock using an additional two assets in this exhaustible resource,
the well-known Solow-Stiglitz is essentially an intertemporal between the returns from holding of the resource and those from amount of capital. Since there are open economy, capital and the this dynamic efficiency condition
255
Resource
extraction, international trade and production with the resource: A. A. Cecen
requires that the returns to these two assets be equal along the optimal path, net of depreciation. Given these necessary conditions, we can proceed to analyse the optimal behaviour ofthe economy under different specifications of the relative market price of the resource. Constant
resource price: pt = p
If the relative price of the resource is constant over time, ie pr = p, the possibility of the opening of trade depends on the value of &, p and 3LZtat t = 0. If: &(O)j
= I,(O)
or, from Equation fi(0)
(12) (10)
= P
then a portion of the extracted resource will be exported in exchange for other foreign investmentconsumption goods for a certain period of time. Since Equation ( 12) compares simply the exogenously given constant price with the marginal product of the resource ,f2 at time t = 0, it follows that when the marginal export revenue, @, is less than the marginal product of the resource in domestic production, ie p <.f*(O), the economy will not extract its resource for export purposes. This situation is basically the violation of the static efficiency condition expressed by Equation (IO), which shows that Ri = 0 when E.r(O)p < j.,(O). Consequently the resource economy will behave, from the very beginning, like a closed economy scrutinized by Dasgupta and Heal [2]. Moreover, considering Equation (7a) together with Equation (8b), we can write, ni,p = L2(0)eP’ if R; > 0
(13)
As t-+cC, the shadow price of the resource, A,, will go to infinity as shown by Equation (8b) in Equation ( 13). Yet, the left-hand side of Equation ( 13) does not tend to infinity as t-+cm because from Equation (5) we have U( C,) = A1 and the assumption of U’( C,) -+ co as C, -+ 0 guarantees that C, > 0 along the optimal path. Hence at a certain point in time, say t = t* the economy will switch to Equation (7b), ie Ai,p < & = for i.,(O)e”’ with RI = 0. Thus at t = t*, extraction export will stop. Therefore: Proposition I. If the exogenously given relative resource price is constant over time and if trade opens, resource export cannot last forever: there will be finite time t* < co when the economy will stop extracting the resource for export and will continue to extract the resource solely for domestic production purposes.
[2]. As is demonstrated in their paper, the essentiality of the resource for production (hence its exhaustion in finite time or infinity) is determined by the magnitude of the elasticity of substitution between capital and the resource, as well as the behaviour of the marginal product of the resource input. If positive output is possible in the absence of the resource input (ie F( K,, L,, 0) > 0) and if the marginal product of the resource is bounded from above as R,-+m 0, then it will be optimal for the economy to deplete the remaining stock of the resource in finite time and continue to produce only with the two traditional inputs, labour and capital. On the other hand, if either of these conditions fails, which is our case since we assumed that F( K,, L,, 0) = 0, then asymptotic depletion is optimal. Exponentially
When the exogenously given resource price increases exponentially at a constant rate & following the same logic above, it is easy to see that the interior solution Equation (7a) can be sustained forever since as t + co, from Equations (7a) and (8b) together with pt = p,,e@, Equation (13) will hold. Hence: Proposition II. If the relative resource price is exponentially increasing at a constant rate, resource extraction both for domestic production and export will continue forever as the resource is asymptotically exhausted.
Summary and conclusion This paper strived to analyse the optimal behaviour of a small resource economy that extracts its exhaustible resource both for domestic production and export purposes. We demonstrated that when the exogenously given relative resource price is constant over time, if trade opens, resource extraction for export would come to an end in finite time. After the economy stops exporting the resource, its optimal growth will be determined simultaneously by the elasticity of substitution between capital and the resource input and the dynamic behaviour of the marginal product of the resource, as depicted by Dasgupta and Heal 123. On the other hand, if the resource price has an exponential trend, we showed that resource extraction would continue forever to support domestic production and the export sector, as the resource is asymptotically depleted.
References 1
For t > t*, the optimal behaviour of the economy will be similar to the one analysed by Dasgupta and Heal
256
increasing resource price: pt = poePf
J. Aarrestad, ‘Optimal savings and exhaustible resource in an open economy’, Journal of Economic Theory, 1978, pp 163-179.
extraction
ENERGY
ECONOMICS
October
199 1
Resource extraction, 2
3
4
P. Dasgupta and G. Heal, ‘Optimal depletion of exhaustible resources’, Review L$ Economic Studies, Symposium, 1974, pp 3-28. P. Dasgupta, R. Eastwood and G. Heal, ‘Resource management in a trading economy’, Quarterly Journal of Economics, 1978, pp 297-306. M. C. Kemp and H. Suzuki, ‘International trade with a wasting but possibly replenishible resource’, International Economic Review, 1975, pp 712-732.
ENERGY
ECONOMICS
October
199 1
international 5
6
7
trade and production
with the resource:
A. A. Cecen
N. V. Long, ‘International borrowing for resource extraction’, International Economic Review, 1974. pp 168-183. M. Moussavian, ‘Growth rates with an exhaustible resource and home goods’, Journal of International Economics, 1985, pp 281-299. N. Vousden, ‘International trade and exhaustible resources: a theoretical model’, International Economic Ret’iew, 1976, pp 149-167.
257