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Global carbon cycle and the optimal time path of a carbon tax.

The existing models of fossil fuel consumption with carbon accumulation imply that the optimal time path of carbon tax is either hump-shaped or monotonically decreasing. These models specify the decay of atmospheric carbon as a constant rate of total concentration. We extend this specification to more accurately reflect the global carbon cycle models of chmatologists and show that this extension changes the basic economic properties of the optimal carbon tax. Our analysis reveals that the optimal carbon tax may as well be constant through time, increase monotonicafly, or have a U-shape. In addition, optimal resource extraction may have an open-close-open cycle.

1. Introduction

Taxation of [CO.sub.2], the principal greenhouse gas emitted from burning of fossil fuels, has been often proposed by economists as a means to control global warming. And in the past few years, the proposal has been the focus of intense debates in policymaking on both sides of the Atlantic Ocean. However, even where policymakers have been persuaded to adopt carbon taxation as a preferred policy instrument, still they have been concerned not only about the initial level of the tax but, perhaps even more, about its time profile too. Such a concern is reflected, for example, in the European Community's proposed carbon tax which starts at $3 per barrel of oil equivalent and increases by a barrel in each subsequent year until year 2000.

Economists, concern about the time profile of carbon tax arises from the facts that fossil fuels are an exhaustible resource and that global warming, being a consequence of carbon accumulation in the atmosphere, is a dynamic (stock) extcmality problem. As such, a carbon tax should be so designed, both in level and in time profile, as to bring about socially desirable paths of fossil fuel consumption and carbon accumulation. Of course, policymakers, worry about political acceptability of the tax is an additional source of concem about the time profile of a carbon tax: a tax which is to rise over time may be more easy to legislate than one which declines throughout time, or vice versa.

It has been in response to these concems that several authors have recently addressed the question of the optimal time path of a carbon tax to control global warming (see Farzin, 1996; Kverndokk, 1994; Sinclair, 1994; and Ulph and Ulph, 1994 among others)(1). These studies, however, differ in their conclusions about the shape of the time path of carbon tax. Thus, whereas Sinclair 1994 argues that the tax should decline over time, Ulph and Ulph (1994) and Kverndokk (1994) argue that it should initially rise over time when the initial stock of accumulated carbon is small, but fall later on when the stock of fossil fuels nears exhaustion. This difference is due to Sinclair's inclusion of production technology and his assumption that it is the percentage reduction in the stock of fossil fuels which determines the growth rate of carbon accumulation. Leaving these differences aside, the time paths of carbon tax found in these studies are limited to those which either are humpshaped (i.e. first rising and then decliningj or decline throughout.(2)

One essential feature shared by most of the above cited studies, however, is that they describe carbon accumulation by a linear differential equation which indicates a constant rate of decay of the carbon stock. This directly implies that carbon accumulation is reversible and that in the long run as the stock of fossil fuels are exhausted and their consumption flow ceases, the stock of accumulated carbon must approach zero. Although this specification may be helpful in gaining some initial insight into the problem, it significantly deviates from the climatologists' models of carbon concentration.

According to IPCC (1990, p. 8-9), the decay of the excess [CO.sub.2] in the atmosphere does not follow a simple exponential curve, and therefore a single linear differential equation cannot describe the accumulation process. From the atmosphere [CO.sub.2] is taken up by oceans and within a short time scale the [CO.sub.2] is mainly taken up by the surface layer of oceans; but within longer time scale of up to several hundred years the principal uptake is by the layers in deep oceans. Climatologists describe the interaction of atmospheric carbon and different layers of the oceans using models which have a multi-time-scale structure. Increase of carbon concentration in the oceans imply that within a time scale of several thousands years the atmospheric concentration will not actually return to its original value. Instead, it reaches a new equilibrium level where about 13-18% of the total amount of [CO.sub.2] emitted will remain in the atmosphere (for details see Maier-raimer and Hasselmann, 1987, and their references). Maier-Raimer and Hasselmann (1987) show that the development of atmospheric carbon concentration and its decay can be approximated by a linear model where the total atmospheric stock is artificially, divided into several substocks with different rates of decay. This structure reflects the internal dynamics of different layers in the oceans and the long term capability of oceans to remove carbon dioxide from the atmosphere.

Given that the climatologists' models of carbon concentration differ in these important respects from the highly stylized models used thus far by economists, we investigate the question of whether the existing economic insights about the optimal time paths of resource consumption and carbon tax are sensitive to the specification of carbon accumulation process. To this end, in Section 2 we develop a model of fossil fuel extraction and carbon accumulation with the distinctive feature that the total accumulated carbon in the atmosphere consists of two different carbon stocks: one which never decays and another which decays over time at a constant proportional rate. This is the simplest possible application of the framework presented by Maier-Raimer and Hasselmann (1987). We also formulate our resource extraction model in such a way that in addition to the standard case of exhausting the resource stock in infinite time, it also permits cases where it is optimal not to exhaust the resource stock or to exhaust it in a finite time. In Section 3, we characterize the optimal time paths by considering some special cases of the more general model in Section 2. One of these identifies the optimal stationary-state policy where the paths of carbon accumulation and carbon tax remain constant throughout time. In the other special case the instantaneous objective function satisfies a quadraticlinear form. We solve this specification in closed form and show various qualitative possibilities for optimal paths. In particular, we show that the optlmal carbon tax may increase monotonically or have a U-shaped form. Both of these cases are ruled out in studies cited above. We also characterize the time paths by numerical simulation. In Section 4, the quadratic-linear specification is extended to include four carbon stocks and global mean temperature. We simulate this model numerically and show that, among other properties, the extraction policy may include an open-close-open cycle. Finally, Section 5 gives concluding remarks.

2. Fossil fuels and carbon accumulation: qualitative analysis

2.1. The model

There is a given (fixed) initial stock of fossil fuel [x.sub.o], which is extracted and burnt at the flow rate of q(t) at time t, implying that the resource stock is depleted at the rate of(3)

x = - q, [x.sub.o] [is greater than] 0

The extraction cost is assumed to be given by the total cost function c(q, x) = qC(x), where, as is standard in the exhaustible resource literature,[4] it is assumed that the unit ( = marginal) extraction cost rises as the resource stock is depleted (C'(x) [is less than] 0) and that the function is convex (C" (x) [is greater than or equal to]0) with C(x) [right arrow] c as x [right arrow] 0 and [is less than or equal to] [infinity]

The flow of emitted carbon, which is taken to be proportional to the flow of fossil fuel burnt,[5] adds to the atmospheric stock of carbon, Z. However, in contrast to the existing literature, in the present model the total stock of atmospheric carbon is described by two carbon stocks with the distinguishing characteristic that one of the stock, [z.sub.1], never decays whereas the other one, [z.sub.2], decays over time at the constant percentage rate [alpha] [is greater than], 0. Accumulation of total carbon stock takes place through additions of emitted carbon to these two stocks. So, if we denote by [a.sub.1], the fraction of a unit of emitted carbon which adds to the non-decaying stock and by [a.sub.2] the fraction which adds to the decaying stock, the accumulation of the two stocks can be described by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the initial carbon stocks [z.sub.10] and [z.sub.20] are given, [a.sub.1] + [a.sub.2] = 1 and where

Z = [z.sub.1] + [z.sub.2]

This description of carbon accumulation is a direct application of the approach developed by Maier-Raimer and Hasselmann (1987). However, we have simplified their model by specifying only one decaying carbon stock (instead of four). By raising the mean global temperature, the total accumulated carbon stock inflicts a flow damage cost which is given by a function D(Z). The damage function is assumed to be increasing (D' [is greater than] 0) and strictly convex (D" [is greater than] 0) with D(0) = 0. In Section 4 we extend the model by including four carbon stocks and the global mean temperature.

The principal question we would like to investigate in this section is: how would our description of carbon accumulation alter the characteristics of optimal fossil fuel consumption and optimal carbon tax as noted in the existing literature? To this end, let us assume that the global planner has a utility function U(q), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and 0 [is less than] q [is less than or equal to] [infinity]. Note that p is a choke price. Formally, the planning problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and (1)-(4). Letting [lambda], [[mu].sub.1], [[mu].sub.2] denote respectively the shadow value of a unit of unextracted resource stock (i.e. the scarcity rent) and the shadow prices of carbon stocks [z.sub.1] and [z.sub.2], the current value Hamiltonian is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that we have changed the signs of the shadow prices for carbon stocks to facilitate economic interpretation. The necessary conditions for an interior optimum (q(t) [is greater than or equal to] 0), when it exits, include

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The cost function, qC(x), which is frequently used in resource economics, does not guarantee the concavity of the maximized Hamiltonian. From (6) we obtain q = q(x, [lambda], [[mu].sub.1], [[mu].sub.2]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using q(x, [lambda], [[mu].sub.1], [[mu].sub.2]) we can eliminate the control variable from the Hamiltonian to obtain the maximized Hamiltonian. After eliminating [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], straightforward calculation shows that the maximized Hamiltonian is concave in (x, [z.sub.2]) if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] this may hold only with some specific functional forms. In these cases a solution which satisfies the necessary conditions together with the transversality conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the optimal solution (Seierstad and Sydsaeter, 1987, theorem 14 p. 236). However, when maximized Hamiltonian is not concave we cannot directly use the transversality conditions. In these cases we apply theorem 16 (p. 244) in Seierstad and Sydsaeter (1987). This theorem specifies conditions which guarantee that eqs (10a-c) can be taken as necessary for optimality. Appendix 1 shows that our problem satisfies these conditions given that C'(0), D ([x.sub.o] + [Z.sub.o]) and D'([x.sub.o] + [Z.sub.o]) are finite. When the maximized Hamiltonian fails to be concave, but the above conditions hold and there is a unique candidate satisfying the necessary conditions (including 10a-c), the existence of the optimal solution implies that the unique candidate is the globally optimal solution.

To facilitate the economic interpretation of the necessary conditions, solve the differential eqs (7) and (8) and use the transversality eqs (10b-c) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus for each type of carbon stocks, the shadow externality cost at time t is the discounted stream of marginal damage costs that a unit of carbon accumulation spills over into the future. According to (6), the marginal utility (or equivalently, the competitive price) of the resource should at any time cover its full social marginal cost which consists of the marginal extraction cost, the scarcity rent, and the shadow extemality cost of an additional unit of carbon added to the existing aggregate stock of carbon. Since an additional unit of emissions contributes both to the accumulation of the non-decaying and decaying stocks in the proportions of [a.sub.1] and [a.sub.2], its shadow externality cost, which is denoted by [mu] [Equivalent] [a.sub.1] + [a.sub.2][[mu].sub.2], and presents the optimal carbon tax when the optimal policy is decentralized, is a weighted average of the shadow externality costs [[mu].sub.1] and [[mu].sub.2]. Equations (8) and (9) ensure the intertemporal efficiency of carbon accumulation for each stock type in that, by equalizing the benefit from any slight delaying of an additional unit of carbon accumulation with the costs of doing so, they ensure the absence of any gain from intertemporal arbitrage.

2.2. Steady state

It shall prove helpful to establish the behavior of the main variables at the limit as t [right arrow] [infinity]. Let us consider the case where it is not optimal to consume the entire stock of fossil fuel. A solution that approaches a steady state satisfies the transversality conditions (10a-c). From (2)-(4) and (7)-(9) it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where the subscript [Infinity] refers to the steady-state solution. Thus along an optimal solution approaching a steady state, the resource rent, resource consumption, and the decaying portion of carbon concentration approach zero. The steady-state total carbon concentration approaches the level of permanent carbon concentration which in turn equals the fraction [a.sub.1] of the total amount of fossil fuel consumed, plus the initial non-decaying stock, i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because the steady-state carbon concentration remains positive, so do the shadow prices for carbon and the emission tax. Using these steady-state levels and (6), we obtain an equation for the steady state resource stock

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At the optimal steady state the choke price equals the unit (or marginal) extraction cost plus the marginal damage cost of permanent carbon concentration. Because the damage function is convex and the unit extraction cost function is decreasing we can deduce that the lower the steady-state resource stock is, the higher is the choke price, the higher is the rate of discount, the higher is the rate of decay of [z.sub.2], and the lower are the initial stock levels of the irreversible carbon and fossil fuel. Accordingly, the lower the steady-state resource stock is, the higher is the steady-state carbon concentration. Note that eq. (12) shows that the steady state is unique. Appendix 2 proves that the steady state is a (local) saddle point. This implies that when the steady state exists our optimality candidate is the (unique) path converging toward it. If there does not exist a solution for eq. (12) with a non-negative resource stock level, it is optimal to deplete the resource stock in finite or infinite time (for details, see Appendix 3).

2.3. Carbon tax time paths in simplified cases

Of particular interest is the time path of total carbon stock which determines the paths of the shadow costs of carbon stocks and therefore the shape of the optimal time path of carbon tax [mu]. Obviously, at the level of generality at which we are analyzing the problem it is not possible to derive these paths analytically. If we assume that [a.sub.1], = 0 and [z.sub.10], = 0 we obtain, as a special case, the model with only the decaying stock of carbon. We first state the basic characteristics of the optimal carbon accumulation and the corresponding carbon tax path in this simpler case for comparison.

We show that if carbon accumulation is described by the decaying stock only, the time path for the emission tax cannot first decrease and later increase. This implies that the time path for emission tax is either monotonically decreasing or is humpshaped. Differentiating (6) with respect to time, assuming [a.sub.1] = 0, [a.sub.2] = 1, and denoting [Mu.sub.2] = [Mu] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The initial level of emission tax must be positive and by the transversality condition (10c) the tax converges to zero as t [right arrow] [infinity]. This implies that if there exists a period of time where the emission tax first decreases and then increases there must be two moments of time where [Mu] = 0. Let us denote these moments by [T.sub.1] and [T.sub.2] so that [T.sub.1] [is less than] [T.sub.2]. At the first turning point [Mu]([T.sub.1] = 0. By eq. (9) this implies that [Mu] = - D"(Z)[z.sub.2]. At the first turning point, the emission tax switches from a decreasing phase to an increasing phase, i.e. [Mu] switches from being negative to being positive. This implies that [Mu]([T.sub.1] = - D" [z([T.sub.1][z.sub.2] ([T.sub.1]) [is less than] 0 and that [z.sub.2] ([T.sub.1] <. 0. By analogous arguments at the second turning point [Mu]([T.sub.2]) = -D" [z([T.sub.2])] [is less than 0. Between [T.sub.1] and [T.sub.2] there must exist a moment, say T, when [z.sub.2] = 0. Because q(T) = [alpha]z(T) [is less than] [alpha][z.sub.2]([T.sub.2]) [is less than] q([T.sub.2]) the level of q mustincrease between T and [T.sub.2]. However, between T and [T.sub.2] we have [Mu] [is greater than] 0 which implies by (13) that extraction must decrease. This is a contradiction and rules out phases where the emission tax first decreases and then increases.(6) Thus we have ruled out the possibility that the carbon tax has a U-shape form. This implies that the time path may either be humpshaped or monotonically decreasing. When [z.sub.20] = 0 eq. (9) immediately shows that [Mu](0) [is greater than] 0 implying that the tax time path is humpshaped when the initial carbon concentration is low enough. On the other hand, when [z.sub.20] is high enough, we obtain [Mu](0) [is greater than] 0, so that the carbon tax must decrease monotonically toward zero. We note that eq. (13) obtains the same form even if C(x) 0. Since the other part of the proof is also independent of the extraction costs, the same result can be obtained without extraction costs as well.

The properties of the time paths of extraction and pollution are more difficult to prove. However, in the case of linear-quadratic forms the various possibilities are derived in Tahvonen (1995). A particular simple case follows if carbon accumulation is described by one non-decaying stock, i.e. [a.sub.2] = [z.sub.20] = 0. In this case the carbon stock is a linear function of the resource stock implying that the model has mathematically only one state variable. If p - C([x.sub.0]) - D([z.sub.10] + [x.sub.10]/[Delta] [is less than or equal to] 0, there exists a unique steady state and the resource stock is not exhausted in finite time. Along the saddle point path resource extraction decreases monotonically toward zero implying that both the carbon stock and emission tax increase monotonically toward their steady state levels (see Farzin, 1995).

We next consider the dynamic behavior of the model with two pollution stocks. Due to the complexity of this model we approach the problem by special cases with exact functional forms.

3. Special cases with closed form solutions

3.1. A stationary policy

We begin by exploring the possibility of a stationary-state policy where the optimal paths of the total carbon stock, hence carbon tax, and resource scarcity rent all remain stationary throughout, that is, an optimal policy characterized by Z(t) = Z, [Mu](t) = [Mu], [lambda](t) = [lambda] t [epsilon] [0, [infinity]. The interest in such a policy may, among other reasons, arise from simplicity, and therefore practicability, of the optimal tax rules that it implies for controlling [CO.sub.2] emissions. By (2), (3), and (4) and recalling that [a.sub.1] + [a.sub.2] = 1

Z = [z.sub.1] + [z.sub.2] = q - [alpha][z.sub.2] (14)

or, using (4) and [z.sub.1] = [a.sub.1]([x.sub.0] - x) + [z.sub.10]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The constancy of the total carbon stock Z = 0 requies by (15) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assuming U'(q) [right arrow] [infinity] as q [right arrow] 0 it must hold along an optimal extraction policy that q [right arrow] 0 and x [right arrow] 0 as t [right arrow] [infinity] (see Appendix 3). This implies that the only stationary path that is consistent with the resource stock constraint is the one for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (17a) in (8) and (9), the constant time paths of the shadow costs of the carbon stocks and the optimal carbon tax are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, in this special case the optimal carbon tax will be constant and at a level which can easily be calculated from the known initial values of the parameters. What is particularly important to note is that the possibility of a constant optimal path of carbon tax is a distinct feature of the present model as it is ruled out in the previous models where only a decaying carbon is considered. This is readily seen by noting that when [z.sub.1] = 0, the constancy of the total carbon stock, Z = [z.sub.2] = 0, requires by (3) a constant extraction path q(t) = q = [alpha][z.sub.2]. But, this implies the exhaustion of the stock in finite time T = [a.sub.2][x.sub.0]/[alpha][z.sub.2], in which case for all t [is greater than] T, q(t) = 0 and by (3), [z.sub.2] [is less than] 0, thus contradicting [z.sub.2] = 0 Vt [epsilon] [0, [infinity]) . We have therefore the following proposition. Proposition I A constant optimal path of the carbon tax is possible when there are both decaying and nondecaying carbon stocks, it is not possible when only the decaying stock is present.

Next substituting from (1) for x(t) in (17a) yields q/q = - [alpha][a.sub.1], which, upon solving for q(t) and noting from (17a) that q(0) = [alpha][a.sub.1][x.sub.0], gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, the stationary optimal policy requires that always a constant percentage of the resource stock be extracted, or, equivalently, the extraction rate should decline at the constant percentage rate of [alpha][a.sub/1]. Finally, a constant optimal time path of the scarcity rent requires that [lambda] = 0 Vt [epsiolon] [0, [infinity], so that one must have or upon differentiating the RHS and using (1)

Denoting [sigma](x) = -xC"(x)/C'(x) the magnitude of the elasticity of C'(x), condition (21) for the constancy of the time path of [lambda](t) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

But noting from (17a) that along the optimal path of carbon accumulation one has q/q = - q/x = (-[alpha][a.sub.1]), it follows that along that path the optimal scarcity rent will also remain constant if and only if [sigma](x0 is constant and equal unity. Formally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 2 A stationary optimal path Z = [[Mu].sub.1] = [[Mu].sub.2] = [Mu] = [lambda] = 0 is possible if and only if the extraction cost function c(q, x) = qC(x) takes the special form for which C'(x) has unitary elasticity.

Furthermore, it is easy to verify that a semi log-linear cost function C(x) = 1n(m/x), where m is a positive constant, exhibits such a property. Finally, the solution is consistent with optimality condition (6) when the utility function satisfies the property U'(q) = 1n(n/q), where [n.sub.1] is a positive constant. Substituting C'(x) = -1/x from this function and q = [alpha][a.sub.1]x from (17a) into (7) yields the optimal stationary value of the scarcity rent as [lambda] = [alpha][a.sub.1]/[delta].(7) Thus, if, for whatever reason, the private owners of fossil fuels were too myopic to allow for the exhaustibility and therefore the scarcity rent of fossil fuel stock, then the planner can correct this by simply levying a fixed tax of [alpha][a.sub.1][delta] on each unit of the resource extracted (see Farzin, 1995).

If the initial state is off the stationary path, explicit solutions for these functional forms are not possible. However, it is likely that the stationary path has a turnpike property, i.e. the other solutions with the initial states off the stationary solution will converge toward this path when t [right arrow] [infinity] (cf stationary paths in growth theory). Given this turnpike property, the stationary solutions summarize a major part of the optimal dynamics of this specification. Although the existence of the stationary solution with constant carbon tax requires specific functional forms, this finding clearly indicates that extending the carbon cycle model changes the possible time profile of the emission tax.

3.2 Quadratic-linear functional forms

In this section we apply the following functional forms: U(q) pq - [bq.sup.2], D(Z) [dz.sup.2], C(x) [c.sub.1] - [c.sub.2]x, where p, b, d, [c.sub.1] [c.sub.2] are positive constants.(8) Let us eliminate [z.sub.1] by using [z.sub.2] = [a.sub.1](x.sub.0] - x) + [z.sub.10] and denote the combined shadow price for the resource stock and non-decaying carbon stock by [psi], i.e. [psi] = [lambda] = [[Mu].sub.1]. Necessary condition (6) implies: q = ([c.sub.2]x - [[mu].sub.2][a.sub.2] + p - [c.sub.1] - [psi])/ 2b q(x, [[Mu].sub.2], [psi]). The Modified Hamiltonian Dynamic System (MHDS) can now be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is a system of four linear first-order differential equations. Its negative characteristic roots9 can be solved by using Theorem 1 in Dockner (1985)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Delta] is the determinant of the Jacobian matrix of the MHDS and equals

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It is possible to show that [[Omega].sup.2] - 4[Delta] 0. Together, these imply that the steady state. when it exists, is a saddle point and that the characteristic roots are real (Dockner, 1985, theorem 3).

Given that it is not optimal to deplete the stock in finite time (i.e. there exists a positive stock satisfying eq. (12)) the solution for the decaying pollution stock can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [A.sub.1], [A.sub.2], are the constants to be determined by the initial `stock levels. Using the equation for [z.sub.2] we obtain the time path for the resource use

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integration and taking into account the steady-state resource stock level yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The equations for x and [z.sub.2] must satisfy the initial conditions x(0) = [x.sub.0] and [z.sub.2](0) = [z.sub.20]. This yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (25), (27), and [lambda], [[Mu].sub.1] and [[mu].sub.2] we obtain rent and the costate variables for the pollution stocks

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that in (30) we have denoted the zero rate of decay of [z.sub1] by [x.sub.1](=0). Finally, the steady state q = [z.sub.2] = x = [z.sub.1] = [z.sub.2] = [lambda] = [[mu].sub.1] = [[mu].sub.2] = 0 determines [[varLambda (Math/Science)].sub.[infinity]] = 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Again in (32b) [alpha].sub.1] = 0. We next characterize the solution paths in two cases i.e. when the initial carbon concentration is low and when it is high.

Proposition 3 When [z.sub.10] and [z.sub.10] are low enough, the time paths for optimal extraction and the resource rent are always declining and the paths for the total carbon stock and the carbon tax are either monotonically increasing or first increasing but later decreasing. For proof, see Appendix 4.

The proof shows that the time path for [z.sub.1] is monotonically increasing and for [z.sub.2] it is hump-shaped. The fact that total carbon concentration may have either of these functional forms is proved by simply varying the relative magnitudes of [a.sub.1] and [a.sub.3]. However, we will show by numerical simulation that these two cases can be generated as well, for example, by varying the steepness of the demand function. We emphasize that only the hump-shaped time paths for carbon concentration and carbon tax are possible when the model consists only of the decaying carbon stock.

The optimal policy for the case where the carbon concentration has already accumulated to some high level can be summarized as follows:

Proposition 4 When [z.SUB.20] is high enough the initial level of extraction first increases but later converges to zero. The paths for total carbon concentration and carbon tax are either monotonically declining or U-shaped. For proof, see Appendix 5.

To interpret this result, notice that because [z.sub.1] increases forever there is a monotonically increasing element in marginal damage and thus also in the path for the emission tax. The other element in the emission tax is due to [z.sub.2]. If [z.sub.20] is high this element is initially high. If the initial state is an outcome of uncontrolled carbon accumulation the optimal level of fossil fuels consumption may be initially very low compared to [[Alpha]z.sub.20]. As a consequence the level of total carbon stock may first decrease although in the long run the irreversible accumulation of [z.sub.1] may dominate the time paths for atmospheric carbon and emission tax. Thus it is possible that the time paths obtain a U-shape.

We note that the differences of the above results compared to those of Ulph and Ulph (1994) follows from our description of [CO.sub.2] accumulation dynamics and not from our inclusion of stock dependent extraction costs. Inspection of the proofs for Propositions 3 and 4 shows that, given that the exhaustion of the resource stock in finite time is non-optimal, the same qualitative results follow when [c.sub.1] = [c.sub.2] = 0. However, in this case the resource rent will be zero (eq. (29)). This is an implication of the non-decaying component of the carbon stock.

The properties of optimal paths depend, not only on the initial values and relative magnitudes of [a.sub.1] and [a.sub.2] but also on the steepness of the demand function. Parameter b does not determine the steady state (eqs (31-32b)). It, however, determines the optimal extraction level and the speed of convergence toward the steady state. This and the other properties of the model are characterized by the numerical examples presented in Figs 1a-d. Figure 1a presents the phase diagram for the model when there is only the decaying carbon stock. As proved analytically, low initial pollution levels imply that the carbon stock level is first increasing although it later converges toward zero. Higher initial pollution levels imply monotonically decreasing pollution. Figure 1b presents a phase diagram when the model includes both the decaying and irreversible carbon stocks. In addition, it is assumed that the demand function is relatively flat (b is low) which implies faster convergence than would otherwise be the case. The dotted lines define the relevant region in the phase diagram. We have assumed that there exist ultimately 3,300 units of natural resource. The lower bound is defined by z = [a.sub.1] (3,300 - x), i.e. along this line [z.sub.2] = 0 but an amount of carbon equal to [a.sub.1 (3,300) - x) exits as the non-decaying carbon stock. The higher bound is defined by z = [a.sub.1] (3,300 - x) + [a.sub.2] (3,300 - x). Figure 1b shows that except for non-zero steady state carbon stock there are no qualitative differences between Figs 1a and 1b. Figure 1c presents optimal trajectories when there are two carbon stocks and the demand function is assumed to be steeper than in Fig. 1b. In contrast to Figs 1a and 1b, a low initial level of carbon implies monotonically increasing stock while a high initial level implies a trajectory which increases toward the steady state. Finally Fig. 1d presents the time paths of emission tax for the same solutions as depicted in Fig. 1c. As shown, the optimal time path either increases monotonically or first decreases but later increases.

It can be asked whether some form of the time paths may be more probable than the others. Within this model it is clear that if the optimal carbon taxation policy is implemented in a situation where carbon concentration equals the pre-industrial level, (here zero) the time path for the carbon stock will either increase forever or have the humpshaped form. The monotonically increasing form follows if the short run atmospheric capacity for [CO.sub.2] decay is high enough compared to human capacity to use fossil fuels. However, it seems to be more likely that the regime of optimal carbon taxation will follow a period of zero carbon taxation, with the consequence that the initial carbon concentration is above the pre-industrial level. This may then lead to monotonically decreasing or U-shape carbon tax paths. U-shape paths are more likely the higher is the rate of fossil fuel consumption during the phase of zero carbon taxation.

4. Extended carbon cycle model with global mean temperature

In this section we extend the carbon cycle model to include four stocks with different rates of decay. This would give more accurate description of the global carbon cycle (see Maier-Raimer and Hasselmann, 1987). In addition, we take into account that atmospheric carbon concentration may not directly enter the damage function. Instead, increases in carbon concentration may drive the global mean temperature away from the pre-industrial state and the difference between present and pre-industrial global mean temperature may be taken as an index of global climate change. We denote this difference by T ([degrees] C). Because the climate system has been adapted to historical carbon concentration and temperature levels there is a memory term which forces the temperature back to the pre-industrial level. Accordingly, an additional differential equation relates the carbon concentration to changes in the global mean temperature. We denote the memory in the temperature equation by [[Alpha].sub.5] T where [[Alpha].sub.5] is a positive constant. The term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represents linearized radiative forcing (see IPCC, 1990, p. 41). These parameters can be estimated from empirical data or from the output of Global Circulation Models (see Tahvonen et al., 1994 or Tahvonen, 1994). With the above extensions, the problem of optimal fossil fuel consumption is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[Alpha].sub.1] [is greater than] 0 i = 2, 3, 4. The system given by eq. (35) specifies the carbon cycle model of Maier-Raimer and Hasselmann (1987) with the only difference that they have five carbon stocks. Equation (36) specifies the relationship between the carbon concentration and the change of temperature. Because of the complexity of this model we do not aim to provide a detailed description of its solution. Rather we will show how this extension may alter some of the basic qualitative properties of the extraction path of exhaustible resources.

Denote the co-state variables for different carbon stocks and temperature by [[Mu].sub.i], i = 1, 2, 3, 4, 5. Because the objective function is quadratic and all constraints are linear this problem can be solved by standard methods (see e.g. Kirk, 1970, p. 209). The MHDS of this problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([is greater than] 0) gives the level of resource consumption that maximizes the Hamiltonian. The MHDS is a non-homogenous system of constant coefficient differential equations. After the elimination of [z.sub.1], using [z.sub.1] = [a.sub.1] ([x.sub.0] - x) + [z.sub.10], the characteristic equation for the Jacobian matrix of the MHDS is a tenth degree polynomial. The characteristic roots can be solved numerically. Solutions which approach a steady state satisfy the infinite horizon transversality conditions. To ensure the stability of the steady state this solution can include only negative characteristic roots (or roots with negative real parts). The assumption that unit extraction costs finally exceeds the choke price, i.e. [c.sub.1] [is greater than] p, is sufficient (but not necessary) for the existence of a steady state with positive resurce stock levels. Denote by [n.sub.j](t) the jth variable of the problem, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The interior solutions i.e. the solutions where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be given in the following form: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [[Gamma].j] is the eigenvector of variable j, [r.sub.i] is a characteristic root, [A.sub.i] is the vector of coefficients to be solved by the initial levels of the state variables and [n.sub.j[[infinity]] is the stady state value of variable j.

Numerical simulation shows that using various parameter values the optimal time paths for the emission tax and carbon stocks may have similar qualitative properties as in the specification of Section 3. In spite of this, the time paths for the optimal resource extraction and resource rent do not necessary follow similar forms as in Sections 2 and 3. This is demonstrated in Figs 2a-b. It is assumed that initially the carbon concentration-temperature system equals the pre-industrial state, i.e. the initial values of carbon stocks and temperature are zero.(11) As Figs 2a and 2b show, the time paths for the emission tax and (total) carbon stock are humpshaped (solid lines). Also the temperature time path satisfies the same form. However, the time paths for the resource use and resource rent first decrease close to zero but after that they have a humpshaped form (Fig. 2a). This form suggest that there may be parameter values where the optimal extraction level is temporarily zero. The possibility of this case is demonstrated by the solution depicted by the dotted line in Fig. 2a. It is computed by assuming that the demand function is flatter (i.e. b( = 0.5) is lower) than in the case represented by the solid line (b = 1). Because the level of extraction is temporarily negative the dotted line is not the correct optimal solution. It, however, suggest that in this case the correct optimal solution consists of three phases. The optimal level of extraction is first positive, then temporarily zero, and finally has the humpshaped form.

Because the level of damage becomes significant after a delay, it may be optimal to extract resources rapidly in the beginning. This causes the carbon concentration and the carbon tax to rise rapidly, thereby slowing down the optimal extraction rate. A period of low extraction rates slows down the rate of carbon concentration, which in turn causes a delayed decrease in temperature. This lowers the level of optimal carbon tax which in turn gives rise to the possibility of a second period of optimal fossil fuel consumption. This extraction profile should be compared to the broken lines which give the outcome of optimal extraction when the carbon tax is set equal to zero (i.e. the Hotelling outcome). The Hotelling extraction path is monotonically decreasing as depicted in Fig. 2a. Notice from Fig. 2b that the Hotelling outcome leads to higher maximum carbon concentration and temperature levels than the optimal solution. Accordingly, the level of the remaining resource stock will be ultimately lower with zero carbon tax.

In exhaustible resource models with accumulating pollution the extraction profile is normally found to be either monotonically decreasing or first increasing and later decreasing. As we showed in sections 2 and 3, high initial pollution levels may give rise to situations where optimal fossil fuel consumption stays at zero until the pollution stock and its shadow cost has decreased from high initial levels. To our knowledge models where the optimal extraction rate of non-renewable resources may endogenously follow an open-close-open cycle are rare in the exhaustible resource literature.(12)

5. Conclusions

The shape of the long-run time path of the carbon tax is an important question from the policy point of view. The classical Hotelling model can be extended for analyzing the time path for the carbon tax and how an accumulating pollution problem like increasing atmospheric [CO.sub.2] alters the economic logic of the optimal exhaustible resource consumption in general. The previous models combining fossil fuels extraction and accumulation of atmospheric [CO.sub.2] have specified carbon accumulation by a single equation with constant rate of decay. Although this specification may be useful for obtaining initial insight to this problem, there is no doubt that it calls for generalizations.

If the exhaustible model contains a single decaying pollution stock the time path for the carbon tax is either decreasing or has an inverted U-shape. If the single stock does not decay the tax increases monotonically. Our first specification includes both a decaying and non-decaying stocks with the implication that all above mentioned time paths are possible. However, in addition to these, carbon tax may as well be constant or have a U-shape. Which of these possibilities will be realized depends, for example, on the initial conditions and on whether the use of fossil fuels has began without pollution control.

A model with four carbon stocks, global mean temperature, and remaining fossil fuels as state variables became too complex for detailed analytical investigation. However, our numerical computations suggest that the carbon tax path may follow any of the cases described above. More surprisingly, we found that the optimal use of fossil fuels may consist of an open-close-open cycle. Together these findings show that the basic properties of resource extraction, and optimal carbon tax in particular, are rather sensitive to the submodel describing the accumulation of atmospheric [CO.sub.2].

REFERENCES

Dockner, E. (1985). `Local Stability Analysis in Optimal Control Problems with Two State Variables', in G. Feichtinger (ed.), Optimal Control Theory and Economic Analysis, 2, North Holland, Amsterdam.

Farzin, Y. H. (1992). `The Time Path of Scarcity Rent in the Theory of Exhaustible Resources', Economic Journal, 102, 813-30.

Farzin, Y. H. (1995). `Optimal Timing of Global Warming and its Control Policy', Fondazione Eni Enrico Mattei, Working Papers No. 86-95.

Farzin, Y. H. (1996). `Optimal Pricing of Environmental and Natural Resource Use with Stock Externalities', Journal of Public Economics, (Forthcoming).

IPCC (International Panel of Climate Change) (1990). Climate Change: the IPCC Scientific Assessment, Cambridge University Press, Cambridge.

Kirk, D. (1970). Optimal Control Theory: An Introduction, Prentice-Hall, Englewood Cliffs, NJ.

Kverndokk, S. (1994). `Depletion of Fossil Fuels and the Impact of Global Warming', Discussion Paper No. 187, Statistics of Norway.

Maier-Raimer, E. and Hasselmann, K. (1987). `Transport and Storage of [CO.sub.2] in the Ocean -- an Inorganic Ocean-circulation Carbon Cycle Model' Climate Dynamics, 2, 63-90.

Schulze, W. (1974) `The Optimal Use of Non-renewable Resources: The Theory of Extraction', Journal of Environmental Economics and Management, 1, 53-73.

Seierstad, K. and Sydsaeter, A. (1987). Optimal Control Theory with Economic Applications, North Holland, Amsterdam.

Sinclair, P. (1994). `On the Trend of Fossil Fuel Taxation', Oxford Economic Papers, 46, 869-77.

Tahvonen, O. (1989). On the Dynamics of Renewable Resource Harvesting and Optimal Pollution Control, Ph.D. dissertation, Publications of Helsinki School of Economics, No. A67, Helsinki.

Tahvonen, O. (1994). `[CO.sub.2] Abatement as a Differential Game', European Journal of Political Economy, 10, 685-05.

Tahvonen, O., (1995). `Trade with Polluting Nonrenewable Resources', Journal of Environmental Economics and Management, 30, 1-17.

Tahvonen, O., von Storch, H., and von Storch, J. (1994). `Economic Efficiency of [CO.sub.2] Reduction Programs', Climate Research, 4, 127-41.

Ulph, A. and Ulph, D. (1994). The Optimal Path of a Carbon Tax, Oxford Economic Papers, 46, 857-68.

APPENDIX 1

1. The necessity of the transversality conditions 10a-c

To apply theorem 16 in Seierstad and Sydsaeter (1987 p. 224) we first note that because C(x) and D(Z) are continuous with continuous first derivatives our problem satisfies the regularity conditions. Define q by U'(q) - C([x.sub.o] = 0. Our assumptions on U(q) and C(x) imply that q must be finite. By (6) it follows that q [is greater than] q(*) where q(*) is the level of q along an optimality candidate. This guarantees that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dt is bounded from above. Given D([Z.sub.0]) is finite, a solution where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that the criteria functional is bounded from below, so that it is bounded from below with any optimality candidate. The fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] dt is bounded follows from the finite amount of the initial resource stock. Next we must find non-negative numbers A, B, a, b, k, and b [greater than] k such that the following growth conditions are satisfied for all t [equal to or greater than] 0 and for all x, [z.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, parameter values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and k = 0 satisfy this requirement implying that the transversality conditions 10a-c are necessary for optimality.

APPENDIX 2

1. The (local) saddle point property of the steady state

Because [z.sub.1] = [a.sub.1]([x.sub.0] - x), + [z.sub.10] the model includes mathematically only two state variables. Using conditions (1)-(4) and (6)-(9) we can write the MHDS as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is derived from (6). The determinant of the Jacobian matrix of this system is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies by the saddle point theorem in Tahvonen (1989) that the steady state is a (local) saddle point.

APPENDIX 3

1. Various cases when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.1. When [lim.sub.q[right arrow]o] U`(q) = [infinity] the resource stock is exhausted in infinite time

In this case no steady state exists and x must converge to zero in finite or infinite time. The former possibility would require that i approached infinity at finite time which contradicts (7). Thus we are left with the case where x converges to zero asymptotically. Along this path eq. (6) must be satisfied which implies that we must have [varLambda (Math/Science)] (t) [right arrow] [infinity] as t [right arrow] [infinity]

1.2. When [lim.sub.q[right arrow]o] U`(q) is finite the resource stock is exhausted in finite time By eq. (6) the path for optimal resource extraction must be continuous. This implies that at the moment when the resource stock is used up, say at T, extraction must equal zero and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Assume that [varLambda(Math/Science)](T) = 0. From (11a,b) it follows that [[Mu.sub.1](t) [less than] 0 and [[Mi].sub.2](t) [less than] 0 for t [greater than ] T. This would imply that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Because (t) [greater than] 0 yields a contradiction with q(t) = 0 for t [equal to or greater than] T, it follows that [varLambda (Math/Science)](T) [greater than] 0. In fact, q(t) = 0 for t [greater than] T, entails [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that during (T, [infinity]), [Mu](t) will be declining at rates less than [varDelta(Math/Science)]. Next, (7) implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.e. the present value of resource rent remains constant after T. Because x(t) = 0 t [equal to or greater than] T this solution satisfies the infinite horizon transversality condition (10). Conditions x(T) = 0, p = c(0) + [varLambda (Math/Science)](T) + [Mu](T) will then enable us to completely solve for the optimal values including those of T and q(0).

APPENDIX 4

Proof of Proposition 3

When [z.sub.20] = 0 eq. (25) implies that [A.sub.i] = - [A.sub.2]. Because [r.sub.1] [less than] [r.sub.2] [less than] 0 (see eq. (24)), [A.sub.2] [less than] 0 and eq. (25) would imply that [z.sub.2] will be negative. Thus when [z.sub.20] = 0 it holds that [A.sub.1] [less than] 0 and [A.sub.2] [greater than] 0. Equation (26) and [A.sub.2] [greater than] 0 imply that [Alpha] + [r.sub.2] [greater than] 0 or otherwise q will become negative. From eq. (28b) we obtain: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] high initial levels of [z.sub.2] would imply that [A.sub.2] is negative which it turn implies by (25) and (26) that [z.sub.2] and q will finally become negative. Thus [Alpha] + [r.sub.1] [is less than] 0. Notice that [Alpha] + [r.sub.1] [is less than] 0 and [Alpha] + [r.sub.2] [is greater than] 0 must hold independently of the initial of [z.sub.2]. Because the optimal trajectories consist two exponential terms the time paths may reach zero slope only once when t [varEpsilon (Math/Science)] [0, [infinity)]. As shown above [z.sub.20] = 0 implies that [A.sub.1] [is less than] 0 and [A.sub.2] [is greater than] 0. Now (25) shows that [z.sub.2](t) first increases but later converges toward zero. From (25) we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we obtain that q(0) [is less than] 0. This directly implies that extraction is monotonically decreasing for all t [equal to or greater than] 0. Note from (29) that [Alpha] + [r.sub.1] [is less than] 0 and [Alpha] + [r.sub.2] [is greater than] 0 imply that the resource rent satisfies the same qualitative form. The properties of optimal extraction imply directly that [z.sub.1] [is greater than] 0, [z.sub.1] [is less than] 0 for all t [is greater than] 0. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the time path for z must be increasing forever when [a.sub.2] is low enough and first increasing but later decreasing when [a.sub.1] is low enough. We now turn to the shadow prices for the pollution stocks.

When Z [is greater than] 0 for all t [varEpsilon (Math/Science)] [0, [infinity]) eqs (8) and (9) directly show that the carbon tax is monotonically increasing. From (30) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By using the fact [r.sub.1] [is less than] [r.sub.2] it follows that [Mu](0) is positive, i.e. both shadow prices are initially increasing given any non-negative level of [a.sub.1]. When [a.sub.1] [right arrow] 0 it follows that [Z.sub.1[infinity]] [right arrow] 0 and [[Mu].sub.[infinity]] [right arrow] 0. In this case the emission tax is initially increasing but later it decreases toward zero.

APPENDIX 5

1. Proof of Proposition 4

From (26) and (28a,b) we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.e. the higher is [z.sub.20] the lower is the initial level of extraction. This implies that when [z.sub.20] is high enough extraction must initially be zero. Equation (26) shows that when q(0) = 0 it holds that q(0) [greater than] 0. Thus when [z.sub.20] is high enough the level of optimal extraction must first increase although it later decreases toward zero.

Using (25) (27) and [z.sub.1] = [a.sub.1([x.sub.o - x]) + [z.sub.10] the time derivative of the total carbon stock can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix 4 shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implying that when [z.sub.20] is high enough both [A.sub.1] and [A.sub.2] would be positive. If we let [a.sub.1] [right arrow] 0 (A3.1) shows that we must finally have Z [is less than] 0 ?? t [is greater than] 0. By eqs (11c,d) this implies a monotonically declining carbon tax. Equation (30) can be used to write the time derivative of the carbon tax in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In eq. (A.3.2) the coefficient of [e.sup.r1t is negative when [A.sub.1] is positive. From the definitions of [Delta] and [Omega] it follows that b [right arrow] [infinity] implies that [Delta] [right arrow] 0 and [Omega] [right arrow] - [Alpha]([Alpha] + [varDelta (Math/Science)]). By eq. (24) we obtain that [r.sub.2] [right arr from below. When [r.sub.2] is close to zero, the term [1 + [a.sub.1]([Alpha] + [r.sub.2]] in (A3.2) must be negative implying that the coefficient for [e.sup.r2t] will be positive, so that [Mu] [is greater than] 0 when t [right arrow] [infinity]. It is possible to show that when b [right arrow] [infinity] the derivative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] approaches a positive constant. This shows that the initial level of carbon tax may obtain arbitrarily high values when b and [z.sub.20] are high enough. Because [[Mu].sub. [infinity]] is independent of b and [z.sub.20] and [Mu] approaches the steady state level from below, the

(1) Among the earlier works is Schulze (1974).

(2) It should be noted that Farzin (1996) addresses the question of optimal time path of carbon tax to control global warming in a model which differs from the other models cited here by formulating a more general model of resource depletion, allowing for a threshold level of stock pollution. assuming no carbon stock decay, and allowing for emission abatement.

(3) Henceforth time subscripts will be obmitted (excluding some initial and terminal levels) and dots over variables will denote their time derivatives.

(4) The treatment of the resource stock as initially fixed and exogenously given as well as the specification of the extraction cost used here present a special case of the more general model of resource extraction studied in Farzin (1992).

(5) By appropriate choise of units we can normalize the factor of proportionality to unity so that a unit of fuel burnt emits a unit of carbon.

(6) A similar proof is presented in an interesting paper by Kverndokk (1994). His argument is based on a claim that q [is greater than] 0 when t [epsilon] [[T.sub.1], [T.sub.2]] and that this yields a contradiction with [Mu] [is greater than] 0. This is, however, a confusion. In addition, ruling out U-shape paths for the emission tax does not imply that the similar shape is ruled out for the pollution stock.

(7) Using the functional forms of this section it follows by (6) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implying that the maximixed Hamiltonian is concave.

(8) A special case of this specification with [a.sub.1] = 0 and [z.sub.10] = 0 is studied in Tahvonen (1995).

(9) Positive characteristic roots can be neglected by the transversality conditions (10a-c).

(10) For the definition of [Omega] see Appendix 2.

(11) The other parameter values are: [x.sub.o] = 3,000, a = 1,000, b = 0.5 or 1, [c.sub.1] = 1,100, [c.sub.2] = 1/3. d = 0.1, [varDelta (Math/Science)] = 0.05, [a.sub.1] = 0.15, [a.sub.2] = 0.3, [a.sub.3] = 0.3, [a.sub.4] = 0.25, [[Alpha].sub.2] = 0.005, [[Alpha].sub.3] = 0.01, [[Alpha].sub.4] = 0.0167, [Mu] = 0.005, [[Alpha].sub.5] = 0.02.

(12) One exception is the finite horizon model by Schulze (1974) where extraction may reach zero level instantaneously.
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Author:Farzin, Y.H.; Tahvonen, O.
Publication:Oxford Economic Papers
Date:Oct 1, 1996
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