- Email: [email protected]
A model of optimal depletion of renewable resources
JOURNAL
OF ECONOMIC
A Model
THEORY
of Optimal
12, 315-324 (1976)
Depletion
of Renewable
Resources
ANTHONY A. SAMPSON University of Sheffield, Shefield SIO ZTN, England Received January 2, 1975; revised July 23, 1975
1. INTRODUCTION The basic issue in resource economics is the question of how far renewable and nondepletable resources can be substituted for depletable, nonrenewable resources. Projections like those of Meadows [6] have been criticized by Nordhaus [7] and Beckerman [l] for ignoring the effects of price changes on depletion rates. Daly [3] suggests that recycling of depletable resources, and increasing use of nondepletable energy sources, may permit higher levels of welfare than are enjoyed at present. In the model presented below, the state of technique in obtaining energy from resources, and the stock of energy yielding resources, are controlled over time by a level of energy consumption and a measure of search for higher thermal efficiency. The criterion is a function of consumption over a planning horizon, plus the present value of steady state consumption possibilities at a terminal time. A complete solution of the model proved impossible, but we could generate a realistic example of the model, for which a complete solution was possible. The model suggests that along an optimal path, the level of search should fall over time, and that the optimal rate of energy consumption should rise and then fall, if the initial.state of technique in converting resources into energy is primitive in relation to some optimal steady state level of technique. This optimistic model is akin to earlier golden age models of economic growth.
2. THE MODEL A realistic model of optimal resource depletion and inventive activity would involve physical capital, pollution, resources, knowledge and population as state variables. The study of optimal trajectories with IZ state variables, by the methods of modem control theory, involves 2n differential equations, with mixed initial and terminal boundary conditions. 315 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
316
ANTHONY
A. SAMPSON
The main analytical device is the phase diagram, which is only applicable to simplified models (see [5] for examples of this problem). Even in the simplified model presented below, a complete solution is possible only for the special case, discussed in Section 4. We define the following variables, at time t: Q(t), E(t), X(t), R(t), M(t), Y(t),
rate of homogeneous output; energy consumption; level of search activity; stock of natural resources; state of technique; public and private consumption
and the following
of output;
functions:
production function, Q = f(E); thermal efficiency function, g(M); cost of search function, C(X). R(t) is a stock of energy yielding resources, assumed homogeneous and measured in “barrels.” In that improvements in technique alter the energy available from different resources by differing ratios, this involves an index number problem, which we ignore. Search for higher thermal efficiency X(t) is measured in man-hours per year, and comprises exploration for new energy sources and investment (physical and research) in new techniques which use less energy per unit of output, as well as the search for higher thermal efficiency. A reduction in the energy content of output is assumed to be equivalent to using fewer resources per unit of energy, at a constant energy content per unit of output. The state of technique M(t), measured in man-hours, is how much search has occurred in the past. In that new workers must be trained in best practice techniques, and energy conversion devices maintained, we assume that a state of technique M requires search at a level 6M to maintain that state. Hence, i@=X--8M (1) The state of technique M determines, via the thermal efficiency function g(M), how many units of energy, measured in therms, can be obtained per barrel. We assume that g(M) > 0 for A4 > m”, g’(M)
> 0, g”(M)
< 0, g’(mO) = Ml, g’(m)
= 0
where m” is “very small” and Ml is “very large.” If E therms per year are generated, then E/g(M) units of resource are consumed.
OPTIMAL
DEPLETION
OF RENEWABLE
317
RESOURCES
Energy consumption E(t) is the only argument in the production function Q =f(E). A constant, fully employed labor force is implied, technical progress is captured by g(M), and the absence of capital stock in E(t) implies a one-commodity economy, with capital consumption permitted, and with a continuously optimal capital stock determined by the myopic rule. We assume that f(E) > 0 for E > e”,f’(E)
> 0, f”(E)
< O,f’(eO) = EO,f’(co)
= 0
(3)
where again e” is “very small” and ED is “very large.” Resource consumption E/g(M), or more precisely the services of a stock of resources, differs from resource depletion, in that some resources may be renewable and/or nondepletable. Fossil fuels are depletable, and on any subgeological time-scale, nonrenewable. Depletable sources such as wood or sewage are renewable, and given some high price for oil, gas and coal, trees would be planted for burning. Tides, winds, hydroelectric, geothermal, thermonuclear, and solar sources of power are nondepletable in that an installation can be operated indefinitely without extraction of any substance from the earth’s crust. Depletion rates of various types of resources would be l? = -E/g(M)
R=o II = h(n) - E/g(m)
for a depletable, non-renewable resource; for a nondepletable resource; for a depletable, renewable resource, where iz is some measure of renewal activity.
The search for higher thermal efficiency includes attempts to develop nondepletable and renewable sources of energy, which would allow indefinite consumption of energy and output. To capture features common to all three types of resource discussed above, we adopt as our resource depletion function a simple amalgam of the three processes ri = p - E/g(m).
(4)
Hence, /3 is a measure of the (natural or man-made) renewal powers of the stock of energy-yielding resources. If search X takes, in part, the form of increasing /3, we assume that its effect is via g(M). The model presented below relies crucially on the assumption that /I > 0. The level of search X(t) has a cost, in terms of immediate consumption foregone, of C(X), and consumption of net output Q is therefore f(E) - C(X). We assume that C(0) = 0, C(X) > 0 for X > 0, C’(X) > 0, C”(X) > 0.
(5)
318
ANTHONY
A. SAMPSON
An increasing marginal cost of search, in terms of consumption forgone, implies an interior optimum with search and consumption positive almost everywhere along an optimal path. A constant marginal cost of search would imply a bang-bang solution for search and consumption.
3. OPTIMAL
PATHS
Given a state of technique M, the function
is, in view of (4), that level of output sustainable indefinitely, further resource depletion. Properties (2) and (3) imply cfr(M) > 0 for M > m’, F(M)
> 0, F(M)
< 0, +(m’)
without
= #JO, @(co)
;7;
where m’ is “very small” and #’ is “very large.” Since a state of technique M can only be maintained by a level of search of 6M, at a cost in consumption of C(6M), we define S(M) = $(M) - C(6M)
(8)
as the steady-state consumption possibility associated with M. Properties (5) and (7) imply that S(M) reaches a maximum at a unique steady-state nir, given by $‘(a) = SC’@&%). Such a state is analogous to the golden age (see [8], where the marginal product of capital equals the sum of the rates of depreciation and population growth. (This anology is not accidental. I am indebted to a referee for pointing out that the model can be interpreted in the standard production function sense, with M a stock of capital, X investment, C(X) the cost of converting output to capital, and Q = f( g(m)@ - R)) the production function.) Steady-state consumption S(M) does not depend on the stock of resources R, and for a while consumption might exceed the golden age level. Optimal resource depletion in our model implies exhausting the stock of depletable nonrenewable resources to finance consumption and research into better methods of using renewable and nondepletable sources of energy. The objective is to select time-paths for consumption E(t) and search X(t), so as to maximize J=
J0
= e-‘“{f(~)
-
C(x)}
dz + (e-‘T/r)i+(M)
-
C@M)LT
(9)
OPTIMAL
DEPLETION
OF RENEWABLE
319
RESOURCES
where the rates of change of the state variables M(t) and R(t) are given by (1) and (4), and subject to nonnegativity requirements and initial conditions X >, 0, E >, 0, R 3 0, M >, 0, R(0) = R”, M(0)
= MO
(10)
(We omit the time-index where to do so would not lead to ambiguity.) The criterion J is the present value of consumption over the interval (0, T), with Tunspecifield, plus the present value of the steady-state consumption possibilities associated with the terminal state of technique M(T), where I’ is the social rate of time preference. In what follows, we can ignore the nonnegativity constraints on X, E, and M. Let TVand e-rtX(t) be Lagrange multipliers associated with the constraints R(T) > 0, R(t) 3 0, and let e-rtw(t) and ~‘~y(t) be auxiliary variables (shadow prices) associated with M(t) and R(t). Form the augmented Hamiltonian (see [4] H = e-W(E)
- C(X) + WV - SM) + Y@ - (E/g(M)))
+ AR). (11)
The auxiliary variables must obey the following differential equations: G = (r + 8) w - ( y&‘(M)/g(M)2),
(12)
j = ry - A,
(13)
and the boundary conditions NT) = WN#(M)
- =‘@Wh=,
,
y(T) = eTTp. Maximizing
(14) (15)
H by choice of controls X and E implies that
f’(E*)
= (Y/~(M)),
c’(x*)
(16)
= w
(17)
where E* and X* denote values along optimal time-paths. The nonnegativity constraints on R(t) imply that R(t) h(t) = 0, R(T) p = 0.
(18)
The terminal date T is not specified, and a value T that maximizes J is given by the following transversality condition (see [22]): f@*) - C(x*) + WV* - SW + Y@ - (E*/g(M))) =
o
it+T
*
+ AR - +(M) + C(6M)
(19)
320
ANTHONY
A.
SAMPSON
A complete solution of the above model, using phase diagrams, eluded the author, since four differential equations are involved. We discuss a partial solution in the rest of this section, and give an example, which can be solved completely, in Section 4. Since R(T) does not appear as an argument in J, an optimal path would imply depletion of resources at time T. (R(T) > 0 would imply p = 0, R(t) > 0 for t < T would imply A(t) = 0, which from (13) and (15) would imply y(t) = 0; hence, from (16) f’(E*) = 0.) Hence R(t) > 0 for t < T, R(T) = 0; hence, from (15) and (16) j = ry, y(t) = ertp.
(20)
That the shadow price of a scarce resource should rise exponentially over time is well-known (see [9]). Since R(T) = 0, E* < /3g(M) must hold at time T, and the absence of R(T) from the expression for J implies E* = ,&(M)t=,
.
(21)
From (6), (12) (16). and (21) we have z.i~= (r + 6) w - $‘(M)lt=T
,
(22)
and the transversality condition (19) simplifies, in view of (6), (16) and (18) to w(x* - SM) - c(x*) + C(SM) = 0. (23) We can prove that the terminal conditions (14) and (23) have a unique solution at the stationary equilibrium G(T) = h’(T) = 0. From (14) dw/dM
= (I/r){q(M)
in view of (5) and (7). Substituting with respect to X*, dM/dX*
= C”(X*)(X*
- PC”(SM)] C’(X*)
t=T < 0
= w in (23) and differentiating
- 8M)/G{C’(X*)
- C’(6M):
> 0
in view of (5), and since dw/dX* > 0, dw/dM > 0 along the locus of (23). Hence, the loci of (14) and (23) can have only one point of intersection. If iPI = 0 then X* = 6M from (I), hence (23) is satisfied. If ik? = zir = 0 then w = $‘(M)/(r + 8) = C’@M), and hence (14) is satisfied. So at time T, the state of technique A4 attains a modified golden age equilibrium M*, given by +‘(M*) = (r + 8) C’@M*). If r > 0 then
OPTIMAL
DEPLETION
OF
RENEWABLE
321
RESOURCES
M* < lti. The approach path to this equilibrium, given by (12), is difficult to analyze, since the ti locus in w - M space depends on the behaviors of E” and y. Further analysis would require phase diagrams of more than two dimensions. It is, fortunately, easy to construct a realistic example which avoids this problem, and this is done in the next section.
4. We choose the following
EXAMPLE
functional
forms for g(M), f(E)
and C(X):
g(M) = a log, M, f(E) = b log, E,
(25)
C(X) = cx2.
(26)
(24)
Apart from having marginal products that diminish more rapidly than for most production functions (i.e., Ef’(E) is constant), these forms appear intuitively reasonable, and satisfy conditions (2), (3), and (5). We assume that a and b are small enough to ensure that M > 1, E > 1 everywhere. From (16) (24) and (25) we have E* = ab log M/y;
(27~
ti = (r + 6) w - (b/M log M).
(28)
hence, from (12)
From (I), (17) and (26) we have iI& = (w/2c)
- 6M.
(29)
From (4), (20) (24), and (27) we have I? = ,!I - (b+/p); hence, R(t) = R” - (b/rp) + Pt + (be++/rp),
(30)
and from (21) we have p = e-rTb//3.
(31)
The peculiar forms adopted for g(M) and f(E) allow us to solve two separate sets of differential equations independently. Equations (28) and (29) determine w and M, and (30) and (31) allow us to solve for p and T. We know that G(T) = 88(r) = 0; hence, the solution for w and M is given in the phase diagram of Fig. 1. (The monotonic form of the ti = 0 locus depends on the functional forms adopted for g(M) andf(E).
322
ANTHONY
A.
FIGURE
SAMPSON
1
The only possible approach paths to the modified golden age equilibrium M* (point P) are the dotted arrowed curves in sectors I and III. We shall only discuss the case where M(0) = MO < M*. Here w(f) falls over time; hence, X*(t) falls. The final value of M* is, from (28) and (29), given by M*2 log M* = b/2c&r + 8).
(32)
From (20) and (27) we have dE*/dt = (ab/y){X*/M)
- 6 - r log M}, d2E*/dt2 < 0.
Since ik’ > 0, there exists a critical value of M(0) at which dE*/dt > 0. Hence if the initial state of technique is very low, an optimal path for energy consumption will rise, and then fall (since X*(T) = 6M(T)) as T is approached. If E* does rise at first, consumption Y = f(E) - C(X) will also rise, since dX*/dt < 0. Consumption will, however, fall as T is approached. To see this dY -dt-
--S-rlogM/
and since X*(T)
-2cX[(r+S)X-
2cMfogMi,
= 6M*, and using (32), dYjdt ltzT = -rb
< 0.
We now determine T and CL.Since R(T) = 0, from (30) and (31), the optimal terminal date T is the solution to R” = (j?/r)(eFT - 1 - rT} = /3((rT2/2!)
+ (r2T3/3!) + a..},
from which aT/ar 1ROcon3t < ‘,
aTPRo I,.const > 0.
(33)
OPTIMAL
DEPLETION OF RENEWABLE RESOURCES
323
Hence, the lower the social rate of time preference, and the larger the initial stock of resources, the longer the resources should last. Since erT > (-) 1 + rT for rT > (=) 0, so long as R” > 0, for any T > 0 (however large), there exists an r > 0 such that (33) holds. Also for any r > 0 (however small), there exists a T > 0 such that (33) holds. Hence, as r gets arbitrarily small, T gets arbitrarily large.
5. CONCLUSIONS
The assumption that energy-yielding resourcesare renewable, or that a large stock of nondepletable resources are available, is crucial to the model. It allows us to resurrect, in a different form, the earlier golden age models. The lower the social rate of time preference, the higher the terminal state of technique, and the longer the period over which resources are depleted. The counter-intuitive statement that energy use and consumption may rise at first applies only once the economy has moved onto the optimal path. These levels of consumption may be lower than the levels prevailing before the economy moved onto its optimal trajectory. The empirical question is whether renewable resources(backstop technology) can ever be as cheap (in terms of human labor per therm) to use as the depletable resources relied on today. Nothing has been said in the paper about population, although the assumption that there is a maximum sustainable level of total output implies that in the long run consumption per head must fall unlessa stationary level of population is reached. ACKNOWLEDGMENT I should like to
my thanks to a referee for many valuable comments.
express
REFERENCES 1. W.
BECKERMAN,
Econ.
Papers
Economists, Scientists and the environmental 24 (1972),
catastrophe, O.uford
327-344.
2. A. E. BRYSON AND Y. C. Ho, “Applied Optimal Control,” Blaisdell, Waltham, Massachusetts, 1969. 3. H. E. DALY, “The Economics of the Steady State,” pp. 15-21, A.E.R. Papers and Proceedings, May 1974. 4. M. I. KAMIEN AND N. L. SCHWARTZ, Sufficient conditions in optimal control theory, J. Econ. Theory 3 (1971), 207-214. 5. E. KEELER, M. SPENCE, AND R. ZECKHAUSER, The optimal control of polution, J. Econ. Theory 4 (1972), 19-34. 6. D. H. 612/12/z-9
MEADOWS
et
al., “The
Limits
to Growth,”
New
York,
1972.
324
ANTHONY A. SAMPSON
7. W. D. NORDHAUS, World dynamics: Measurement without data, Econ. J. 83 (1973), 1156-1183. 8. E. S. PHELPS, “Golden Rules of Economic Growth,” W. W. Norton, New York, 1966. 9.
R. M. SOLOW, “The Economics of Resources and the Resources of Economics,” pp. l-14, A.E.R. Papers and Proceedings, May 1974.
OF ECONOMIC
A Model
THEORY
of Optimal
12, 315-324 (1976)
Depletion
of Renewable
Resources
ANTHONY A. SAMPSON University of Sheffield, Shefield SIO ZTN, England Received January 2, 1975; revised July 23, 1975
1. INTRODUCTION The basic issue in resource economics is the question of how far renewable and nondepletable resources can be substituted for depletable, nonrenewable resources. Projections like those of Meadows [6] have been criticized by Nordhaus [7] and Beckerman [l] for ignoring the effects of price changes on depletion rates. Daly [3] suggests that recycling of depletable resources, and increasing use of nondepletable energy sources, may permit higher levels of welfare than are enjoyed at present. In the model presented below, the state of technique in obtaining energy from resources, and the stock of energy yielding resources, are controlled over time by a level of energy consumption and a measure of search for higher thermal efficiency. The criterion is a function of consumption over a planning horizon, plus the present value of steady state consumption possibilities at a terminal time. A complete solution of the model proved impossible, but we could generate a realistic example of the model, for which a complete solution was possible. The model suggests that along an optimal path, the level of search should fall over time, and that the optimal rate of energy consumption should rise and then fall, if the initial.state of technique in converting resources into energy is primitive in relation to some optimal steady state level of technique. This optimistic model is akin to earlier golden age models of economic growth.
2. THE MODEL A realistic model of optimal resource depletion and inventive activity would involve physical capital, pollution, resources, knowledge and population as state variables. The study of optimal trajectories with IZ state variables, by the methods of modem control theory, involves 2n differential equations, with mixed initial and terminal boundary conditions. 315 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
316
ANTHONY
A. SAMPSON
The main analytical device is the phase diagram, which is only applicable to simplified models (see [5] for examples of this problem). Even in the simplified model presented below, a complete solution is possible only for the special case, discussed in Section 4. We define the following variables, at time t: Q(t), E(t), X(t), R(t), M(t), Y(t),
rate of homogeneous output; energy consumption; level of search activity; stock of natural resources; state of technique; public and private consumption
and the following
of output;
functions:
production function, Q = f(E); thermal efficiency function, g(M); cost of search function, C(X). R(t) is a stock of energy yielding resources, assumed homogeneous and measured in “barrels.” In that improvements in technique alter the energy available from different resources by differing ratios, this involves an index number problem, which we ignore. Search for higher thermal efficiency X(t) is measured in man-hours per year, and comprises exploration for new energy sources and investment (physical and research) in new techniques which use less energy per unit of output, as well as the search for higher thermal efficiency. A reduction in the energy content of output is assumed to be equivalent to using fewer resources per unit of energy, at a constant energy content per unit of output. The state of technique M(t), measured in man-hours, is how much search has occurred in the past. In that new workers must be trained in best practice techniques, and energy conversion devices maintained, we assume that a state of technique M requires search at a level 6M to maintain that state. Hence, i@=X--8M (1) The state of technique M determines, via the thermal efficiency function g(M), how many units of energy, measured in therms, can be obtained per barrel. We assume that g(M) > 0 for A4 > m”, g’(M)
> 0, g”(M)
< 0, g’(mO) = Ml, g’(m)
= 0
where m” is “very small” and Ml is “very large.” If E therms per year are generated, then E/g(M) units of resource are consumed.
OPTIMAL
DEPLETION
OF RENEWABLE
317
RESOURCES
Energy consumption E(t) is the only argument in the production function Q =f(E). A constant, fully employed labor force is implied, technical progress is captured by g(M), and the absence of capital stock in E(t) implies a one-commodity economy, with capital consumption permitted, and with a continuously optimal capital stock determined by the myopic rule. We assume that f(E) > 0 for E > e”,f’(E)
> 0, f”(E)
< O,f’(eO) = EO,f’(co)
= 0
(3)
where again e” is “very small” and ED is “very large.” Resource consumption E/g(M), or more precisely the services of a stock of resources, differs from resource depletion, in that some resources may be renewable and/or nondepletable. Fossil fuels are depletable, and on any subgeological time-scale, nonrenewable. Depletable sources such as wood or sewage are renewable, and given some high price for oil, gas and coal, trees would be planted for burning. Tides, winds, hydroelectric, geothermal, thermonuclear, and solar sources of power are nondepletable in that an installation can be operated indefinitely without extraction of any substance from the earth’s crust. Depletion rates of various types of resources would be l? = -E/g(M)
R=o II = h(n) - E/g(m)
for a depletable, non-renewable resource; for a nondepletable resource; for a depletable, renewable resource, where iz is some measure of renewal activity.
The search for higher thermal efficiency includes attempts to develop nondepletable and renewable sources of energy, which would allow indefinite consumption of energy and output. To capture features common to all three types of resource discussed above, we adopt as our resource depletion function a simple amalgam of the three processes ri = p - E/g(m).
(4)
Hence, /3 is a measure of the (natural or man-made) renewal powers of the stock of energy-yielding resources. If search X takes, in part, the form of increasing /3, we assume that its effect is via g(M). The model presented below relies crucially on the assumption that /I > 0. The level of search X(t) has a cost, in terms of immediate consumption foregone, of C(X), and consumption of net output Q is therefore f(E) - C(X). We assume that C(0) = 0, C(X) > 0 for X > 0, C’(X) > 0, C”(X) > 0.
(5)
318
ANTHONY
A. SAMPSON
An increasing marginal cost of search, in terms of consumption forgone, implies an interior optimum with search and consumption positive almost everywhere along an optimal path. A constant marginal cost of search would imply a bang-bang solution for search and consumption.
3. OPTIMAL
PATHS
Given a state of technique M, the function
is, in view of (4), that level of output sustainable indefinitely, further resource depletion. Properties (2) and (3) imply cfr(M) > 0 for M > m’, F(M)
> 0, F(M)
< 0, +(m’)
without
= #JO, @(co)
;7;
where m’ is “very small” and #’ is “very large.” Since a state of technique M can only be maintained by a level of search of 6M, at a cost in consumption of C(6M), we define S(M) = $(M) - C(6M)
(8)
as the steady-state consumption possibility associated with M. Properties (5) and (7) imply that S(M) reaches a maximum at a unique steady-state nir, given by $‘(a) = SC’@&%). Such a state is analogous to the golden age (see [8], where the marginal product of capital equals the sum of the rates of depreciation and population growth. (This anology is not accidental. I am indebted to a referee for pointing out that the model can be interpreted in the standard production function sense, with M a stock of capital, X investment, C(X) the cost of converting output to capital, and Q = f( g(m)@ - R)) the production function.) Steady-state consumption S(M) does not depend on the stock of resources R, and for a while consumption might exceed the golden age level. Optimal resource depletion in our model implies exhausting the stock of depletable nonrenewable resources to finance consumption and research into better methods of using renewable and nondepletable sources of energy. The objective is to select time-paths for consumption E(t) and search X(t), so as to maximize J=
J0
= e-‘“{f(~)
-
C(x)}
dz + (e-‘T/r)i+(M)
-
C@M)LT
(9)
OPTIMAL
DEPLETION
OF RENEWABLE
319
RESOURCES
where the rates of change of the state variables M(t) and R(t) are given by (1) and (4), and subject to nonnegativity requirements and initial conditions X >, 0, E >, 0, R 3 0, M >, 0, R(0) = R”, M(0)
= MO
(10)
(We omit the time-index where to do so would not lead to ambiguity.) The criterion J is the present value of consumption over the interval (0, T), with Tunspecifield, plus the present value of the steady-state consumption possibilities associated with the terminal state of technique M(T), where I’ is the social rate of time preference. In what follows, we can ignore the nonnegativity constraints on X, E, and M. Let TVand e-rtX(t) be Lagrange multipliers associated with the constraints R(T) > 0, R(t) 3 0, and let e-rtw(t) and ~‘~y(t) be auxiliary variables (shadow prices) associated with M(t) and R(t). Form the augmented Hamiltonian (see [4] H = e-W(E)
- C(X) + WV - SM) + Y@ - (E/g(M)))
+ AR). (11)
The auxiliary variables must obey the following differential equations: G = (r + 8) w - ( y&‘(M)/g(M)2),
(12)
j = ry - A,
(13)
and the boundary conditions NT) = WN#(M)
- =‘@Wh=,
,
y(T) = eTTp. Maximizing
(14) (15)
H by choice of controls X and E implies that
f’(E*)
= (Y/~(M)),
c’(x*)
(16)
= w
(17)
where E* and X* denote values along optimal time-paths. The nonnegativity constraints on R(t) imply that R(t) h(t) = 0, R(T) p = 0.
(18)
The terminal date T is not specified, and a value T that maximizes J is given by the following transversality condition (see [22]): f@*) - C(x*) + WV* - SW + Y@ - (E*/g(M))) =
o
it+T
*
+ AR - +(M) + C(6M)
(19)
320
ANTHONY
A.
SAMPSON
A complete solution of the above model, using phase diagrams, eluded the author, since four differential equations are involved. We discuss a partial solution in the rest of this section, and give an example, which can be solved completely, in Section 4. Since R(T) does not appear as an argument in J, an optimal path would imply depletion of resources at time T. (R(T) > 0 would imply p = 0, R(t) > 0 for t < T would imply A(t) = 0, which from (13) and (15) would imply y(t) = 0; hence, from (16) f’(E*) = 0.) Hence R(t) > 0 for t < T, R(T) = 0; hence, from (15) and (16) j = ry, y(t) = ertp.
(20)
That the shadow price of a scarce resource should rise exponentially over time is well-known (see [9]). Since R(T) = 0, E* < /3g(M) must hold at time T, and the absence of R(T) from the expression for J implies E* = ,&(M)t=,
.
(21)
From (6), (12) (16). and (21) we have z.i~= (r + 6) w - $‘(M)lt=T
,
(22)
and the transversality condition (19) simplifies, in view of (6), (16) and (18) to w(x* - SM) - c(x*) + C(SM) = 0. (23) We can prove that the terminal conditions (14) and (23) have a unique solution at the stationary equilibrium G(T) = h’(T) = 0. From (14) dw/dM
= (I/r){q(M)
in view of (5) and (7). Substituting with respect to X*, dM/dX*
= C”(X*)(X*
- PC”(SM)] C’(X*)
t=T < 0
= w in (23) and differentiating
- 8M)/G{C’(X*)
- C’(6M):
> 0
in view of (5), and since dw/dX* > 0, dw/dM > 0 along the locus of (23). Hence, the loci of (14) and (23) can have only one point of intersection. If iPI = 0 then X* = 6M from (I), hence (23) is satisfied. If ik? = zir = 0 then w = $‘(M)/(r + 8) = C’@M), and hence (14) is satisfied. So at time T, the state of technique A4 attains a modified golden age equilibrium M*, given by +‘(M*) = (r + 8) C’@M*). If r > 0 then
OPTIMAL
DEPLETION
OF
RENEWABLE
321
RESOURCES
M* < lti. The approach path to this equilibrium, given by (12), is difficult to analyze, since the ti locus in w - M space depends on the behaviors of E” and y. Further analysis would require phase diagrams of more than two dimensions. It is, fortunately, easy to construct a realistic example which avoids this problem, and this is done in the next section.
4. We choose the following
EXAMPLE
functional
forms for g(M), f(E)
and C(X):
g(M) = a log, M, f(E) = b log, E,
(25)
C(X) = cx2.
(26)
(24)
Apart from having marginal products that diminish more rapidly than for most production functions (i.e., Ef’(E) is constant), these forms appear intuitively reasonable, and satisfy conditions (2), (3), and (5). We assume that a and b are small enough to ensure that M > 1, E > 1 everywhere. From (16) (24) and (25) we have E* = ab log M/y;
(27~
ti = (r + 6) w - (b/M log M).
(28)
hence, from (12)
From (I), (17) and (26) we have iI& = (w/2c)
- 6M.
(29)
From (4), (20) (24), and (27) we have I? = ,!I - (b+/p); hence, R(t) = R” - (b/rp) + Pt + (be++/rp),
(30)
and from (21) we have p = e-rTb//3.
(31)
The peculiar forms adopted for g(M) and f(E) allow us to solve two separate sets of differential equations independently. Equations (28) and (29) determine w and M, and (30) and (31) allow us to solve for p and T. We know that G(T) = 88(r) = 0; hence, the solution for w and M is given in the phase diagram of Fig. 1. (The monotonic form of the ti = 0 locus depends on the functional forms adopted for g(M) andf(E).
322
ANTHONY
A.
FIGURE
SAMPSON
1
The only possible approach paths to the modified golden age equilibrium M* (point P) are the dotted arrowed curves in sectors I and III. We shall only discuss the case where M(0) = MO < M*. Here w(f) falls over time; hence, X*(t) falls. The final value of M* is, from (28) and (29), given by M*2 log M* = b/2c&r + 8).
(32)
From (20) and (27) we have dE*/dt = (ab/y){X*/M)
- 6 - r log M}, d2E*/dt2 < 0.
Since ik’ > 0, there exists a critical value of M(0) at which dE*/dt > 0. Hence if the initial state of technique is very low, an optimal path for energy consumption will rise, and then fall (since X*(T) = 6M(T)) as T is approached. If E* does rise at first, consumption Y = f(E) - C(X) will also rise, since dX*/dt < 0. Consumption will, however, fall as T is approached. To see this dY -dt-
--S-rlogM/
and since X*(T)
-2cX[(r+S)X-
2cMfogMi,
= 6M*, and using (32), dYjdt ltzT = -rb
< 0.
We now determine T and CL.Since R(T) = 0, from (30) and (31), the optimal terminal date T is the solution to R” = (j?/r)(eFT - 1 - rT} = /3((rT2/2!)
+ (r2T3/3!) + a..},
from which aT/ar 1ROcon3t < ‘,
aTPRo I,.const > 0.
(33)
OPTIMAL
DEPLETION OF RENEWABLE RESOURCES
323
Hence, the lower the social rate of time preference, and the larger the initial stock of resources, the longer the resources should last. Since erT > (-) 1 + rT for rT > (=) 0, so long as R” > 0, for any T > 0 (however large), there exists an r > 0 such that (33) holds. Also for any r > 0 (however small), there exists a T > 0 such that (33) holds. Hence, as r gets arbitrarily small, T gets arbitrarily large.
5. CONCLUSIONS
The assumption that energy-yielding resourcesare renewable, or that a large stock of nondepletable resources are available, is crucial to the model. It allows us to resurrect, in a different form, the earlier golden age models. The lower the social rate of time preference, the higher the terminal state of technique, and the longer the period over which resources are depleted. The counter-intuitive statement that energy use and consumption may rise at first applies only once the economy has moved onto the optimal path. These levels of consumption may be lower than the levels prevailing before the economy moved onto its optimal trajectory. The empirical question is whether renewable resources(backstop technology) can ever be as cheap (in terms of human labor per therm) to use as the depletable resources relied on today. Nothing has been said in the paper about population, although the assumption that there is a maximum sustainable level of total output implies that in the long run consumption per head must fall unlessa stationary level of population is reached. ACKNOWLEDGMENT I should like to
my thanks to a referee for many valuable comments.
express
REFERENCES 1. W.
BECKERMAN,
Econ.
Papers
Economists, Scientists and the environmental 24 (1972),
catastrophe, O.uford
327-344.
2. A. E. BRYSON AND Y. C. Ho, “Applied Optimal Control,” Blaisdell, Waltham, Massachusetts, 1969. 3. H. E. DALY, “The Economics of the Steady State,” pp. 15-21, A.E.R. Papers and Proceedings, May 1974. 4. M. I. KAMIEN AND N. L. SCHWARTZ, Sufficient conditions in optimal control theory, J. Econ. Theory 3 (1971), 207-214. 5. E. KEELER, M. SPENCE, AND R. ZECKHAUSER, The optimal control of polution, J. Econ. Theory 4 (1972), 19-34. 6. D. H. 612/12/z-9
MEADOWS
et
al., “The
Limits
to Growth,”
New
York,
1972.
324
ANTHONY A. SAMPSON
7. W. D. NORDHAUS, World dynamics: Measurement without data, Econ. J. 83 (1973), 1156-1183. 8. E. S. PHELPS, “Golden Rules of Economic Growth,” W. W. Norton, New York, 1966. 9.
R. M. SOLOW, “The Economics of Resources and the Resources of Economics,” pp. l-14, A.E.R. Papers and Proceedings, May 1974.