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Depreciation rules and value invariance with extractive firms
Journal of Economic Dynamics & Control 26 (2002) 99}116
Depreciation rules and value invariance with extractive "rms John M. Hartwick , Larry Karp, Ngo Van Long * Department of Economics, Queen's University, Kingston, Ontario, Canada K7L 3N6 Department of Agricultural and Resource Economics, University of California, Berkeley, CA 94720-3311, USA Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2T7
Abstract The application of Samuelson's theorem on value invariance to the case of intertemporally optimizing "rms is shown to require a judiciously chosen economic depreciation formula which depends on both current stock and current #ow variables, in order to prevent the "rms from changing their actions in the face of the tax regime. We illustrate by deriving depreciation rules which achieve non-distortingness of actions and value-invariance for resource-extracting "rms. 2002 Elsevier Science B.V. All rights reserved. JEL classixcation: H25; D92
1. Introduction Samuelson (1964) proved the following fundamental theorem of value invariance: &if, and only if, true loss of economic value is permitted as a tax-deductible depreciation expense will the present discounted value of a cash}receipt stream
* Corresponding author. Tel.: 1-514-398-4844; fax: 1-514-398-4938. E-mail address: [email protected] (N.Van Long). Sinn (1987, p. 119) refers to this result as the Johansson}Samuelson theorem, since the idea also appeared in Johansson (1961) in Swedish. Fane (1987) also refers to Johansson (1961,1969). 0165-1889/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 1 9 - 1
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be independent of the rate of tax' (p. 604). Value invariance is important because it is ezciency-enhancing: it is a su$cient condition for neutral business taxation, under the assumption (which we maintain throughout this paper) that the tax does not cause the rate of interest to change. In particular, in the case of natural resources, ownership of a resource deposit should not be dependent on one's marginal tax rate, but on one's comparative advantage of managing it. In general, however, neutrality with respect to investment does not require value invariance: it only requires that the sign of net present value of all investment projects remain unchanged when the tax is imposed. Value invariance is a strong form of neutrality. Many authors have sought weaker (i.e., less demanding) forms of neutrality. In this paper, we pursue the issue of value invariance in the context of taxation of natural resource "rms because this strong form of neutrality does imply extraction path invariance, which is desirable in the sense that distortions are avoided. Admittedly, we are abstracting from the more important but more di$cult question: what is the best set of biases (distortions) to have, given that some sort of distortions is unavoidable? We restrict our attention to the following question: can we be sure that the Samuelsonian economic depreciation will ensure value invariance and extraction path invariance, if "rms try to reduce the tax burden by contemplating deviation from the extraction path which would be optimal under the no-tax scenario? In the analysis of resource extraction "rms, each with a given resource stock (we abstract from capital equipment for simplicity), resource economists are often interested in the e!ects of taxation on the time path of extraction, which may be regarded as disinvestment, since extraction reduces the resource stock. (See, for example, Dasgupta and Heal, 1979; Gaudet and Lasserre, 1986). The purpose of this paper is to explore the invariance issue in the context of extractive "rms. As will be seen below, in this context, value invariance goes hand-in-hand with extraction path invariance to the tax rate. If one interprets each feasible extraction path as an investment project (or, rather, disinvestment project), then the invariance of the "rm's optimal extraction path to the tax rate The rate of interest would not be a!ected by the tax rate if (a) we are dealing with a small open economy that can borrow or lend at the world rate of interest, which the small country takes as exogenously given, or (b) the marginal product of capital in the economy is constant over the relevant range. The assumption that the nominal discount rate is unchanged by the presence of the tax is a standard one, see, for example, Bond and Devereux (1995, p. 59). In this paper we take this partial equilibrium approach. For a model that takes into account the e!ect of taxation on the equilibrium interest rate, see Long and Sinn (1984). The cash #ow tax, for example, is neutral but does not imply value invariance; in this paper, we assume that the cash #ow tax is not available. For example, the tax proposal in Bond and Devereux (1995) achieves neutrality by &imposing no tax on marginal investment projects; revenue is raised by taxing the pure pro"ts or economic rents earned on infra-marginal investment'. (p. 58).
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is a su$cient condition for tax neutrality within the resource sector. It is conceivable that there are tax schemes that ensure value invariance without extraction path invariance, and vice versa. However, in this paper we restrict attention to tax schemes that ensure both sorts of invariance. Dasgupta and Heal (1979, pp. 370}371) show that if depreciation allowance is based on the observed change in the market value of the "rm, then extraction path invariance is obtained. They did not however address the question of the behavior of the extractive "rm seeking to reduce its tax burden. Samuelson's theorem has been shown to apply also to the case of uncertainty (see Richter, 1986; Fane, 1987) and to the case of a time-dependent tax rate (see Lyon, 1990). As in Samuelson's original paper, in these articles it is assumed that the cash-#ow stream is exogenous or remains unperturbed by the tax regime. It might seem at "rst sight that Samuelson's value invariance theorem can be easily applied to the case of an intertemporally optimizing "rm. Consider "rst the no-tax scenario. Take any feasible time path of production and investment, denoted by q(.); the corresponding time path of the value of the "rm (given that it is committed to this production}investment path) can be computed, and hence the corresponding economic depreciation at each date along that path is known. Samuelson's result implies that the introduction of an income tax which permits economic depreciation to be tax-deductible will not change the value of the "rm, if the production}investment path q(.) is kept unchanged. It follows that the best (i.e., value-maximizing) production}investment path under the no-tax scenario, say qH(.), remains the best one under the Samuelsonian tax. This argument relies on the assumptions that market value of a "rm can be observed at any given point, and that such value is independent of possible attempts by the "rm to reduce its tax liability. In this paper, we show, with the "rm running its known stock via extraction taking into account the impact of its chosen paths on its tax liability, that it is not a trivial problem to get the "rm to reproduce, under the Samuelsonian tax scheme, its no-tax paths of control variable and state variable. In other words, value invariance is di$cult to implement because the "rm tends to deviate from its no-tax paths in the face of the tax and the depreciation allowance. We set out a scheme, inspired by the analysis of taxing a resource monopolist by Karp and Livernois (1992) * who did not consider the issue of depreciation allowance and value invariance * which leads to value (and path) invariance. Our result indicates that the scheme is informationally demanding. The authorities need a large amount of knowledge about the "rm in order to set out an appropriate depreciation allowance of the (true) Samuelson form. We work in a partial A referee pointed out that if the market value of a "rm could be observed, then there would exist an obvious ex-post depreciation rule based on the observed value. In practice, many "rms are not listed on the stock exchanges, and for those that are, volatility of observed share prices seems to work against the idea of basing depreciation allowance on observed market value of the "rm.
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equilibrium framework and focus on the response of the "rm to the taxation scheme, as in a game between the taxation authority striving for neutrality with respect to extraction path, and the "rm, striving for maximum after-tax pro"t. The interest rate is taken as given, as would be the case for a small open economy, or for an economy with constant marginal product of capital over a relevant range. Depreciation allowances are a central topic in mineral economics (Heaps and Helliwell, 1985), and we are dealing with the theoretically ideal depletion allowance for extractive "rms with non-durable outputs (as in oil, coal, etc., as distinct from copper, gold, etc.). For an extractive "rm, the decision at each date is one of optimal disinvestment from its current stock of reserves. Though formally optimal investment and disinvestment can be analyzed the same way under Samuelson taxation, the latter turns out to be simpler because the "rm has no incentive to borrow and its single-state variable (its remaining stock) moves in a simple fashion with respect to its single control variable (its current extraction). In an optimal production}investment problem, the possibility of borrowing makes the link between the current production decision (the control), and the state variable (the capital stock of the "rm) complicated because current investment is another control variable for the "rm. Hence our analysis, based on the purely extractive "rm, is in a sense prior to that of the producing-investing "rm. We show that a Markovian depreciation rule would provide the incentive for a "rm not to deviate from its best extraction (disinvestment) path for the no-tax scenario and has the subgame perfectness property: even if a "rm has deviated (say accidentally) at some time, it will not have an incentive to deviate in the future from its best path achievable given the current state. Our invariance procedure can be described as follows: have the tax authority work out the "rm's optimal extraction path under the no-tax scenario and, given this path, have the authority obtain the time path of true economic depreciation, contingent on the state of the "rm. The tax authority then assigns this depreciation allowance to the "rm for its optimization under the Samuelson taxdepreciation allowance scenario. Value of the "rm invariance and extraction As pointed out by a referee, if each level of Samuelson's tax results in a distinct interest rate, then true value invariance will never obtain. But a standard assumption in partial equilibrium analysis is the rate of interest does not change. For example, Bond and Devereux (1995, p. 59) assume that &the current valuation of the entitlement 2is not a!ected by the introduction of the tax. This assumption parallels the requirement under certainty that the nominal discount rate is unchanged by the presence of the tax'. For analysis with extractive "rms, see Dasgupta and Heal (1979), Heaps (1985), and Gaudet and Lasserre (1986). Long and Sinn (1984) focus on extractive "rms and the neutrality question, but in a speci"c general equilibrium framework. The neutral business tax for producing}investing "rms is addressed in Sandmo (1974,1979), Stiglitz (1973, 1976), King (1974, 1975), Harberger (1980), Boadway and Bruce (1984), Fane (1987), Lyon (1990), Bond and Devereux (1995), and others.
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path invariance obtain. The invariance depreciation formula must be tied to the state variable in order to prevent the "rm from reacting to the formula and, roughly speaking, the invariance formula must be derived in the no-tax environment in order that the "rm assigns the correct shadow price to the state variable under the tax regime. We add two new bits to the Samuelson tax invariance literature: (a) we demonstrate that the &true economic depreciation' in the tax depreciation scheme must be derived in the no-tax scenario, and (b) we demonstrate that the depreciation allowance must be &state-based' in order to rule out deviations by the taxed "rm. We also provide a rigorous economic analysis of the concept of a &depletion allowance' for mining economics. Several authors have paid attention to economic depreciation in the context of intertemporal optimization; but they have not dealt with the examination of alternative tax rules that supposedly represent Samuelsonian true economic depreciation, and the possibility of deviation by the "rm. In fact, some authors have chosen to work with a diwerent concept of economic depreciation. Sandmo (1974) de"ned the &true' rates of depreciation as the true rate of physical wear-and-tear rate plus the interest rate, minus the rate of increase in the price of the investment good. Atkinson and Stiglitz (1980, p. 142) seemed to equate Samuelsonian true economic depreciation with the replacement cost of physical wear-and-tear. Howitt and Sinn (1989), by abstracting from natural resources and pure pro"t, dealt with a model with much simpli"ed economic depreciation, as invariance was not a central concern of their paper. None of these papers contemplates the possibility that the "rm may deviate from a path in order to reduce its tax burden. In contrast, we focus precisely on the issue of the "rm trying to reduce its tax burden, given a depreciation allowance rule. We emphasize the possibility that a tax rule which would ensure value invariance if the "rm would commit to a given path may not work if the xrm can deviate. In this respect, our paper is in the same spirit as that of Karp and Livernois (1992), but that paper did not deal with Samuelsonian economic depreciation: the authors were addressing the question of regulating a monopolist so as to achieve the extraction path that a social planner would choose. They reported on alternative taxing procedures, in the case of monopoly, which would lighten the information required by the tax setting authorities. Implementing the Samuelson tax scheme is information intensive. The "rm must "le its cost structure with the tax authorities at time zero. Contingent on its stock, s(t), of remaining reserves, the government calculates a depreciation allowance rate (s(t)) which will result in the "rm extracting the optimal quantity q(t). The rate (s(t)) will change in general as the stock size shrinks. Samuelson (1964) was not explicit about how this tax scheme was to be implemented but the implicit rate in the depreciation allowance would also vary in his model. The analysis in Samuelson (1964) is very short and leaves one with the impression that value invariance and its associated path invariance is easy to implement. Clearly, neutrality is an obvious desideratum of a tax scheme. Our analysis
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suggests that neutrality is not easy to be achieved via legislated &true economic depreciation'. Like many results in optimal taxation, Samuelsonian invariance (value and path) is hard to implement because the tax authority must "ne tune its &rules' for each "rm. The rest of this paper is organized as followed. Section 2 investigates, in a context of a very simple model of a resource extracting "rm, the incentive for the "rm to respond to the various depreciation allowance rules by deviating from its best production}investment path in the no-tax scenario. The correct economic depreciation rule is then derived, and is shown to have the Markovian property. Section 3 generalizes the result. Section 4 o!ers several additional examples.
2. A simple model Consider a resource extracting "rm. The initial stock in the ground is s . The rate of extraction at time t is q(t). Let s(t) be the stock available at time t. Then ds s (t), "!q(t). dt
(1)
The cost of extraction is C(q), where C(0)"0, C'0 and C'0. To simplify, assume that the "rm sells its output in the world market, at an exogenously given price p which is constant over time. The rate of interest is r, a positive constant. It is assumed that r does not change with the tax rate. This is the case if the country is a small open economy facing a given world interest rate. In the absence of tax, the "rm wishes to "nd an extraction path q(t)50 and a terminal date ¹ to maximize the integral of discounted cash #ow:
2 e\PR[pq(t)!C(q(t))] dt (2) subject to (1), and s(0)"s , s(¹)"0. It is well known that the optimal path must satisfy the following Euler equation, called the Hotelling rule, which says that the net price (i.e. price minus marginal cost) must rise exponentially at the rate of interest: 1 d(p!C) "r. p!C dt Since p is constant, this rule reduces to !C(q(t))q (t)"r[p!C(q(t))].
(3)
This di!erential equation and the following conditions: q(¹)"0
(4)
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and
2 q(t) dt"s (5) uniquely determine the optimal time path qH(t) and the optimal exhaustion time ¹H. If we substitute qH(t) and ¹H into integral (2), we obtain the value of the "rm at time t"0, which we denote as JH(0). Similarly, the value of the "rm at time t50 is de"ned as
2H 2H e\PV\R[pqH(x)!C(qH(x))] dx, e\PV\RH(t) dt, (6) R R where H(t) denotes the pro"t at time t. From this, we can compute the rate of change in the value of the "rm JH(t)"
JQ H(t)"rJH(t)!H(t).
(7)
Economic depreciation is de"ned as the negative of this rate of change DH(t),!JQ H(t).
(8)
Now consider the introduction of a tax scheme whereby pro"t, net of some (at present unspeci"ed) depreciation allowance which we denote by D (t), is taxed at O a rate . Interest income is also taxed at this rate, and interest payments are tax deductible. If the xrm does not change its extraction path, then its value at time t in this tax scenario is
2H e\P\OV\R(1!)[pqH(x)!C(qH(x))]#D (x) dx, (9) O R where the subscript in JH(t) indicates that the value is computed in the tax O scenario. Di!erentiate JH(t) with respect to time, we get O JQ H(t)"(1!)rJH(t)!(1!)H(t)!D (t). (10) O O O Clearly, if D (t)"!JQ H(t), then (10) reduces to O O JQ H(t)"rJH(t)!H(t), (11) O O which is identical in form to Eq. (7). This fact, together with JH(¹H)"JH(¹H)"0, implies that JH(t)"JH(t) for all t. And the depreciation O O allowance that permits this invariance is D (t)"!JQ H(t)"DH(t),!JQ H(t), i.e., O O one can use the depreciation in the no-tax scenario as the depreciation in the tax scenario. This is of course just a restatement of Samuelson's invariance result. It serves as a departure point for our main question: would the "rm have an incentive to change its extraction path? JH(t)" O
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Clearly, if the government announces the tax rate and the time path DH(t) of allowance for depreciation, regardless of whether the "rm might change its extraction path, then the "rm would seek to maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#DH(t) dt
(12)
while taking the time path DH(t) as given. The Euler equation for this problem is 1 d(1!)(p!C) "(1!)r, (1!)(p!C) dt which implies !C(q(t))q (t)"(1!)r.
(13)
This equation is di!erent from (3). This shows that the "rm will have an incentive to change its extraction path. The reader would perhaps object to this argument, on the ground that in the objective function (12) we have inserted a depreciation allowance which corresponds to the &best' extraction path (under the no-tax scenario), and not a truly Samuelsonian economic depreciation that corresponds to the "rm's actual extraction path. This point is well taken. We are well aware that in (12), we do not present the Samuelsonian scheme, and that nobody would seriously advocate the depreciation scheme represented in (12). We have chosen to introduce this naive scheme only as a device to start o! some meaningful discussion. The point is, for each feasible time path q(t), one needs a corresponding D (t) so that the Samuelsonian invariance result applies. This O creates a tremendous informational obstacle to the implementation of the Samuelsonian scheme: a "rm must supply to the tax o$ce, at time t, information about its extraction rate for all points of time beyond t. For otherwise one cannot calculate the change in the value of the "rm at time t. And the tax o$ce must verify that such plan is actually followed by the "rm. We refer to this depreciation allowance scheme as B, and the one represented in (12) as A. Scheme B is informationally very expensive. In what follows, we locate a few other schemes that lie between these two extremes. The reader is forewarned that the solution ultimately presented also requires a great deal of information. We have seen that depreciation scheme A does not ensure invariance, because the depreciation allowance DH(t) is independent of the action that the "rm might choose, even though it is derived from the pro"t-maximizing behavior of the "rm in the no-tax scenario. Consider again the resource-extracting "rm in the no-tax scenario. It is well known that the current-value Hamiltonian at time t equals rJH(t). Thus, rJH(t)"HH(t)"(qH(t))! H(t)qH(t),
(14)
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where H(t) is the shadow price of the resource stock, which by the maximum principle, equals the net price H(t)"p!C(qH(t)).
(15)
From (14), (15), (7) and (8), we obtain DH(t)"[p!C(qH(t))]qH(t),
(16)
which says an eminently sensible thing: economic depreciation for the resource extracting "rm is equal to the amount extracted multiplied by the scarcity rent. Note that if the xrm does not change behavior, this depreciation results in no tax on the marginal unit of production: pre-tax cash #ow is p!C(qH(t)) on the last unit of production, which is equal to the marginal depreciation allowance. So it would seem that the following depreciation allowance scheme might ensure Samuelsonian invariance: If the "rm extracts q(t) (which need not be the same as its extraction in the no-tax scenario), the tax authority allows it to deduct the amount [p!C(q(t))]q(t). We call this economic depreciation scheme C. Under this scheme, the "rm would maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#[p!C(q(t))]q(t) dt (17) subject to the usual constraints. However, it is clear that the Euler equation for this problem involves the third derivative C(q). So this scheme would make the "rm choose some extraction path that is di!erent from qH(t), and invariance would not be obtained. To avoid a third derivative in the Euler equation, one might be tempted to use the following modi"ed scheme, called the modixed C economic depreciation scheme: Allow the "rm to deduct [p!C(qH(t))]q(t) where p!C(qH(t)) is evaluated along the optimal path obtained in the no-tax scenario, while q(t) is the actual extraction at t. The "rm would then maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#[p!C(qH(t))]q(t) dt, (18) where [p!C(qH(t))] is treated as a known function of time, and we denote it as
(t). The "rm takes this function as exogenously given, because its choice of the path q(t) does not a!ect this function. The Hamiltonian is H"(1!)[pq(t)!C(q(t))]# (t)q(t)! (t)q(t). O
As can be veri"ed on the examples in Section 4, our proposed solution has the property that, even if the "rm tries to contemplate deviation so as to reduce the tax burden, the depreciation allowance results in no tax on the marginal unit of production: pre-tax cash #ow on the last unit of production is equal to the marginal depreciation allowance. We thank a referee for pointing this out.
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Unfortunately, this yields an Euler equation which is quite di!erent from the one in the no-tax scenario. We move on to another formulation. We are now introducing a scheme that does ensure the Samuelsonian invariance. Let <(s) be the value function of the mining "rm's optimization problem under the no-tax case. More precisely, let
2 <(s)"max e\PV\R[pq(x)!C(q(x))] dx O 2 R subject to s(t)"s,
s "!q
and s(¹)"0. We know that <(s(t))" (t)"p!C(q (t)) for any s(t)50 where q (t) is the optimal extraction rate at time t given that the "rm optimizes from time t with the starting stock s(t), in the no-tax scenario, i.e., q "q(s) (where the super script opt indicates optimal choice, the subscript n refers to the no-tax scenario, and the subscript refers to the situation with Samuelsonian taxation). Let us try the following depreciation allowance formula, which we call the stock-andyow-dependent depreciation scheme, or the E scheme. d<(s(t)) D (t, s(t), q(t))"! "!<(s(t))s (t)"<(s(t))q(t). O dt
(19)
Then, under the tax scenario, the "rm maximizes
2 e\P\OR(1!)[pq(t)!C(q(t))]#<(s(t))q(t) dt (20) subject to the usual constraints. (Note that this is a problem for the extractive "rm under the tax scenario, and <(s(t)) has been obtained for this "rm under the no-tax scenario.) The Hamiltonian for problem (20) is H"(1!)[pq(t)!C(q(t))]#<(s(t))q(t)! (t)q(t) O and we get the necessary conditions H "(1!)(p!C) #<(s)! "0, O O q
(21)
(22)
Q "r(1!) !<(s)q. O O Di!erentiate (22) with respect to t to get
(23)
(1!)C(q)q "<(s)s ! Q . O
(24)
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From (23) and (24), !C(q)q "r , O which can be expressed as
(25)
d [p!C] "r O O dt or as !C(q)q "r[(1!)(p!C)#<(s)].
(26)
But since <(s)"p!C(q(s)), where q(s) is optimal under the no-tax scenario, Eq. (26) becomes !C(q)q "r(1!)(p!C(q)) #(p!C(q)). O This reduces to the Euler equation (or r% rule) for the no-tax extraction problem when C(q)"C(q)). We now provide a formal demonstration that the two extraction paths q(t) and q(s(t)) will be the same, given <(s)"p!C(q(s)), i.e., under the depreciation scheme D (t)" O [p!C(q(s))]q(t). This proof relies on the Hamilton}Jacobi}Bellman (HJB) equation. For the no-tax case, the HJB equation for the resource extracting "rm is r<(s)"max [pq!C(q)!<(s)q]. (27) O Assume that we have found a function <(s) that satis"es the functional equation (27). From this, we can obtain the function <(s)q (for the no-tax case), which we now use as the depreciation allowance in the tax scenario. Then the value function in the tax scenario, denoted by < (s), must satisfy the following O equation: (1!)r< (s)"max [(1!)(pq!C(q))#<(s)q!< (s)q]. O O O
(28)
Clearly this functional equation has as solution < (.)"<(.), because if we O substitute this proposed solution into (28), we will get (1!)r<(s)"max [(1!)(pq!C(q))#(1!)<(s)q] (29) O and <(s) clearly satis"es this functional equation, because it satis"es the HJB equation (27) which is identical to it (since the terms (1!) on both sides cancel each other out). This proof makes clear that the depreciation formula for invariance (or neutrality) must be based on the true economic depreciation for the "rm's stock
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under the no-tax scenario, not true economic depreciation for the "rm's stock under the tax scenario, even though, ex-post, the two are identical. A key insight in this regard is: the shadow price of a unit of stock moves at r% under the no-tax scenario, and at (1!)r% (minus an adjustment term, see (23)) under the tax scenario. The shadow prices are distinct in general but will be the same under the invariance tax scheme. The general result could be stated as follows: the neutral tax scheme results in < (s), the shadow price for stock under the tax O scenario, made the same as <(s), the shadow price for stock under the no-tax scenario. This completes the proof that our stock-and-yow-dependent depreciation formula D"<(s)q ensures the Samuelsonian invariance, even though the modixed C economic depreciation formula, which is very similar to it, does not. Even though <(s)"p!C(q) when q is optimally chosen given s, i.e. when q"q(s), it is incorrect to use [p!C(q)]q as the depreciation formula, because this would not re#ect the fact that the scarcity value depends on the stock level s. (It would be correct, however, to use [p!C(q(s))]q.) The following proposition sum marizes our main result so far: Proposition 2.1. For the resource extraction xrm in our model, the correct depreciation formula that ensures the Samuelsonian invariance is D"<(s)q, or equivalently, D"[p!C(q(s))]q, but not D"[p!C(q)]q. Our result indicates that for the real-world application of the Samuelson tax scheme, quite a lot of information is needed. The tax authority must collect data on each "rm on costs and stock characteristics, and solve the optimal extraction program, under the no-tax scenario, and then work out true economic depreciation, with the time path of the shadow price of the stock expressed as function of the remaining stock. The depreciation allowance, q(t)<(s(t)) is then assigned to the extractive "rm at each date. 3. Generalizations We now generalize our formula to allow for (i) the possible time-dependence of the cost and revenue functions, and (ii) possible natural growth of the resource. We replace the expression pq!C(q) by the more general net cash #ow function N(s, q, t), and we assume that s "f (s, q, t). In the no-tax scenario, the Hamilton}Jacobi}Bellman (HJB) equation for the "rm's optimization problem is
r<(s, t)"max N(s, q, t)# O
<(s, t) <(s, t) f (s, q, t)# s t
.
(30)
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We assume that this functional equation has a solution, which we denote by <(s, t), which is the value of the "rm at time t if the stock at that time is s. We de"ne the economic depreciation as
d<(s, t) <(s, t) D(s, q, t)"! "! dt s
f (s, q, t)!
<(s, t) . t
(31)
Now consider the scenario with a tax tate and some depreciation allowance A(s, q, t). The HJB equation for the "rm is then (1!)r< (s, t) O
< < O f (s, q, t)# O "max (1!)N(s, q, t)#A(s, q, t)# s t O
, (32)
where < (s, t) is the value of the "rm at time t, in the tax scenario, if the stock at O that time is s. What form should A(s, q, t) take to ensure that < (s, t) is the same as <(s, t)? O Proposition 3.1. If the government sets the depreciation allowance rule to be the same as the function describing true economic depreciation in the no-tax scenario, i.e., if it sets
A(s, q, t)"!
<(s, t) <(s, t) f (s, q, t)! , s t
(33)
then Eq. (32) has the solution < (s, t)"<(s, t), i.e., the value of the xrm in the tax O scenario is equal to its value in the no-tax scenario. Proof. Substituting (33) into (32), we see that the resulting functional equation has <(s, t) as solution. (In other words, replacing < (s, t) (and its partial derivaO tives) by <(s, t) (and its partial derivatives) in that functional equation, we obtain an equation that is identical to (30), except for the terms (1!) on both sides, which cancel each other out.) Remark. Proposition 3.1 is also correct if r and are functions of time. It is easy to see that our proof can be modi"ed to accommodate this time-dependence of r and . The case where s and q are vectors can also be easily accommodated. Strictly speaking, the transversality condition lim e\PR<(s, t)"0 must be satis"ed. R
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4. Some examples In this section, we consider some examples that illustrate how the Samuelsonian true economic depreciation should be represented to ensure that the "rm does not deviate from its best path in the no-tax scenario. 4.1. Example 1 Consider a monopolist that costlessly extracts a natural resource. Let q(t) denote the rate of extraction, and s(t) the resource stock. Then s (t)"!q(t).
(34)
The demand function is p"q\?, 0((1.
(35)
The rate of interest is r, a positive constant. In the absence of taxation, the "rm maximizes the present value of the net cash #ow N(t)"p(t)q(t)"q(t)\?, subject to q50 and lim s(t)"0 R
(36)
s(0)"s (given). This problem yields the necessary condition q "!rq, hence q(x)"q(t)e\PV\R ?.
(37)
(38)
Eqs. (34)}(38), imply rs(0) q(0)"
(39)
and hence the optimal extraction rule is rs(t) q(t)" .
(40)
The value of the "rm at time t is therefore
? q(x)\?e\PV\R dx" s(t)\?. r R It is easy to verify that this solution satis"es the HJB equation <(s(t))"
r<(s)"max[q\?!<(s)q], where the maximization is with respect to q50.
(41)
(42)
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Now introduce an income tax at the constant rate . The "rm's after tax net cash #ow at time t is (1!)q(t)\?#D(t),
(43)
where D(t) is the Samuelsonian true economic depreciation:
D(t)"!
? s(t)\?q(t). r
(44)
This depreciation formula involves both stock and #ow, and implicitly takes into account the pure pro"t, which is re#ected in the elasticity of demand parameter in (44). When there is tax on capital income and where economic depreciation is tax deductible, the HJB equation is
? (1!)r< (s)"max (1!)q\?#(1!) s\?q!< (s)q . O O r
(45)
One can verify that the following value function satis"es (3.12):
? < (s)" s\?. O r
(46)
Comparing (46) with (41), we con"rm that the value of the "rm is independent of the tax rate, and the tax has no e!ect on the "rm's optimal extraction path. To conclude this example, we note that the above invariance result can also be veri"ed by using the standard methods of optimal control theory. One would then obtain the following Hamiltonian to be maximized with respect to q
H"(1!)q\?#(1!) with
? s\?q! q r
(47)
Q "(1!)r #(1!)
? s\?\q. r
(48)
The following necessary condition can then be derived after some manipulation:
!(1!)q\?\q "r(1!)q\?#r(1!)
? s\?. r
(49)
Again, setting q"rs/ (and therefore q "rs /"!rs/), we "nd that (49) is indeed satis"ed. 4.2. Example 2 A competitive "rm extracts a natural resource at the cost c per unit. Its net cash #ow at t is N(t)"[p(t)!c]q(t), where the "rm takes p(t) as given. In the
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industry equilibrium with n "rms having identical unit cost c, the price path must obey the Hotelling rule p "r(p!c).
(50)
It is clear that the value of the "rm at time t, when its resource stock is s(t), is <(s(t), t)"(p(t)!c)s(t).
(51)
Indeed, (51) satis"es the HJB equation r<(s, t)"max [(p(t)!c)q!< q#< ], Q R O
(52)
where from (51) < "p(t)!c, < "p (t)s(t)"r(p!c)s(t), where the last equality Q R follows from (50). The "rm's rate of extraction is of course indeterminate because of the constant unit cost assumption. The "rm's economic depreciation is D(t)"!
< < O q# O "max (1!)(p!c)q#(p!c)q!p s! O s t
.
Again, < (s, t)"<(s, t)"(p(t)!c)s solves this functional equation. Value inO variance is again con"rmed.
5. Conclusion We have shown that when the application Samuelson's value invariance theorem to the case where "rms optimize intertemporally requires a judicious representation of economic depreciation so that the "rm would not deviate from the production}investment path that it would choose in the no-tax scenario. While a given path of economic depreciation can be represented in several di!erent ways, only the Markovian representation does not distort incentives. This representation ensures &subgame perfectness'. In this paper, we have restricted attention to the polar cases where the "rm is either a monopolist or a perfectly competitive "rm. An extension to the case of
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oligopoly would not be trivial, because under oligopoly, each "rm's value depends not only on its stock level, but also on the stock levels of its rivals. In that case, a "rm's true economic depreciation would seem to depend on its rivals' actions. This is a challenging research topic that is part of our future research agenda.
References Atkinson, B., Stiglitz, J.E., 1980. Lectures on Public Economics. McGraw-Hill, Maidenhead. Boadway, R., Bruce, N., 1984. A general proposition on the design of a neutral business tax. Journal of Public Economics 24, 231}239. Bond, S.R., Devereux, M.P., 1995. On the design of a neutral business tax under uncertainty. Journal of Public Economics 58, 57}71. Dasgupta, P., Heal, G.M., 1979. Economic Theory of Exhaustible Resources. James Nisbet, New York. Fane, G., 1987. Neutral taxation under uncertainty. Journal of Public Economics 33, 95}105. Gaudet, G., Lasserre, P., 1986. Capital income taxation, depletion allowances and non-renewable resource extraction. Journal of Public Economics 29, 241}253. Harberger, A.C., 1980. Tax neutrality in investment incentives. In: Aaron, H.J., Boskin, M.J. (Eds.), The Economics of Taxation. Brookings Institution, Washington, DC, pp. 299}313. Heaps, T., 1985. The taxation of non-replenishable natural resources revisited. Journal of Environmental Economics and Management 11, 1}14. Heaps, T., Helliwell, J.F., 1985. The taxation of natural resources. In: Auerbach, A.J., Feldstein, M. (Eds.), Handbook of Public Economics, Vol. 1. North Holland, Amsterdam, pp. 427}465. Howitt, P., Sinn, H.-W., 1989. Gradual reforms of capital income taxation. American Economic Review 79, 106}124. Johansson, S.-E., 1961. Skatt, Investering, Vardering.. Norsted & Soner, Stockholm. Johansson, S.-E., 1969. Income taxes and investment decisions. Swedish Journal of Economics 71, 104}110. Karp, L., Livernois, J., 1992. On e$ciency-inducing taxation for a non-renewable resource monopolist. Journal of Public Economics 49, 219}239. King, M.A., 1974. Taxation and the cost of capital. Review of Economic Studies 41, 21}35. King, M.A., 1975. Taxation, corporate "nancial policy, and the cost of capital. Journal of Public Economics 4, 271}279. Long, N.V., Sinn, H.-W., 1984. Optimal taxation and economic depreciation: a general equilibrium model with capital and an exhaustible resource. In: Kemp, M.C., Long, N.V. (Eds.), Essays in the Economics of Exhaustible Resources. North Holland, Amsterdam. Lyon, A.B., 1990. Invariant valuation when tax rates change over time. Journal of Political Economy 98, 433}437. Richter, W.F., 1986. Comprehensive versus neutral income taxation. Mimeo, Universitat Dortmund. Samuelson, P.A., 1964. Tax deductibility of economic depreciation to insure invariant valuation. Journal of Political Economy 72, 604}606. Sandmo, A., 1974, Investment incentives and the corporate income tax. Journal of Political Economy 82, 287}302.
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Sandmo, A., 1979, A note on the neutrality of the cash #ow corporation tax. Economics Letters 4, 173}176. Sinn, H.-W., 1987. Capital Income Taxation and Resource Allocation. Elsevier Science Publishers, Amsterdam. Stiglitz, J.E., 1973. Taxation, corporate "nancial policy, and the cost of capital. Journal of Public Economics 2, 1}34. Stiglitz, J.E., 1976. The corporation tax. Journal of Public Economics 5, 303}311.
Depreciation rules and value invariance with extractive "rms John M. Hartwick , Larry Karp, Ngo Van Long * Department of Economics, Queen's University, Kingston, Ontario, Canada K7L 3N6 Department of Agricultural and Resource Economics, University of California, Berkeley, CA 94720-3311, USA Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2T7
Abstract The application of Samuelson's theorem on value invariance to the case of intertemporally optimizing "rms is shown to require a judiciously chosen economic depreciation formula which depends on both current stock and current #ow variables, in order to prevent the "rms from changing their actions in the face of the tax regime. We illustrate by deriving depreciation rules which achieve non-distortingness of actions and value-invariance for resource-extracting "rms. 2002 Elsevier Science B.V. All rights reserved. JEL classixcation: H25; D92
1. Introduction Samuelson (1964) proved the following fundamental theorem of value invariance: &if, and only if, true loss of economic value is permitted as a tax-deductible depreciation expense will the present discounted value of a cash}receipt stream
* Corresponding author. Tel.: 1-514-398-4844; fax: 1-514-398-4938. E-mail address: [email protected] (N.Van Long). Sinn (1987, p. 119) refers to this result as the Johansson}Samuelson theorem, since the idea also appeared in Johansson (1961) in Swedish. Fane (1987) also refers to Johansson (1961,1969). 0165-1889/02/$ - see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 1 9 - 1
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be independent of the rate of tax' (p. 604). Value invariance is important because it is ezciency-enhancing: it is a su$cient condition for neutral business taxation, under the assumption (which we maintain throughout this paper) that the tax does not cause the rate of interest to change. In particular, in the case of natural resources, ownership of a resource deposit should not be dependent on one's marginal tax rate, but on one's comparative advantage of managing it. In general, however, neutrality with respect to investment does not require value invariance: it only requires that the sign of net present value of all investment projects remain unchanged when the tax is imposed. Value invariance is a strong form of neutrality. Many authors have sought weaker (i.e., less demanding) forms of neutrality. In this paper, we pursue the issue of value invariance in the context of taxation of natural resource "rms because this strong form of neutrality does imply extraction path invariance, which is desirable in the sense that distortions are avoided. Admittedly, we are abstracting from the more important but more di$cult question: what is the best set of biases (distortions) to have, given that some sort of distortions is unavoidable? We restrict our attention to the following question: can we be sure that the Samuelsonian economic depreciation will ensure value invariance and extraction path invariance, if "rms try to reduce the tax burden by contemplating deviation from the extraction path which would be optimal under the no-tax scenario? In the analysis of resource extraction "rms, each with a given resource stock (we abstract from capital equipment for simplicity), resource economists are often interested in the e!ects of taxation on the time path of extraction, which may be regarded as disinvestment, since extraction reduces the resource stock. (See, for example, Dasgupta and Heal, 1979; Gaudet and Lasserre, 1986). The purpose of this paper is to explore the invariance issue in the context of extractive "rms. As will be seen below, in this context, value invariance goes hand-in-hand with extraction path invariance to the tax rate. If one interprets each feasible extraction path as an investment project (or, rather, disinvestment project), then the invariance of the "rm's optimal extraction path to the tax rate The rate of interest would not be a!ected by the tax rate if (a) we are dealing with a small open economy that can borrow or lend at the world rate of interest, which the small country takes as exogenously given, or (b) the marginal product of capital in the economy is constant over the relevant range. The assumption that the nominal discount rate is unchanged by the presence of the tax is a standard one, see, for example, Bond and Devereux (1995, p. 59). In this paper we take this partial equilibrium approach. For a model that takes into account the e!ect of taxation on the equilibrium interest rate, see Long and Sinn (1984). The cash #ow tax, for example, is neutral but does not imply value invariance; in this paper, we assume that the cash #ow tax is not available. For example, the tax proposal in Bond and Devereux (1995) achieves neutrality by &imposing no tax on marginal investment projects; revenue is raised by taxing the pure pro"ts or economic rents earned on infra-marginal investment'. (p. 58).
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is a su$cient condition for tax neutrality within the resource sector. It is conceivable that there are tax schemes that ensure value invariance without extraction path invariance, and vice versa. However, in this paper we restrict attention to tax schemes that ensure both sorts of invariance. Dasgupta and Heal (1979, pp. 370}371) show that if depreciation allowance is based on the observed change in the market value of the "rm, then extraction path invariance is obtained. They did not however address the question of the behavior of the extractive "rm seeking to reduce its tax burden. Samuelson's theorem has been shown to apply also to the case of uncertainty (see Richter, 1986; Fane, 1987) and to the case of a time-dependent tax rate (see Lyon, 1990). As in Samuelson's original paper, in these articles it is assumed that the cash-#ow stream is exogenous or remains unperturbed by the tax regime. It might seem at "rst sight that Samuelson's value invariance theorem can be easily applied to the case of an intertemporally optimizing "rm. Consider "rst the no-tax scenario. Take any feasible time path of production and investment, denoted by q(.); the corresponding time path of the value of the "rm (given that it is committed to this production}investment path) can be computed, and hence the corresponding economic depreciation at each date along that path is known. Samuelson's result implies that the introduction of an income tax which permits economic depreciation to be tax-deductible will not change the value of the "rm, if the production}investment path q(.) is kept unchanged. It follows that the best (i.e., value-maximizing) production}investment path under the no-tax scenario, say qH(.), remains the best one under the Samuelsonian tax. This argument relies on the assumptions that market value of a "rm can be observed at any given point, and that such value is independent of possible attempts by the "rm to reduce its tax liability. In this paper, we show, with the "rm running its known stock via extraction taking into account the impact of its chosen paths on its tax liability, that it is not a trivial problem to get the "rm to reproduce, under the Samuelsonian tax scheme, its no-tax paths of control variable and state variable. In other words, value invariance is di$cult to implement because the "rm tends to deviate from its no-tax paths in the face of the tax and the depreciation allowance. We set out a scheme, inspired by the analysis of taxing a resource monopolist by Karp and Livernois (1992) * who did not consider the issue of depreciation allowance and value invariance * which leads to value (and path) invariance. Our result indicates that the scheme is informationally demanding. The authorities need a large amount of knowledge about the "rm in order to set out an appropriate depreciation allowance of the (true) Samuelson form. We work in a partial A referee pointed out that if the market value of a "rm could be observed, then there would exist an obvious ex-post depreciation rule based on the observed value. In practice, many "rms are not listed on the stock exchanges, and for those that are, volatility of observed share prices seems to work against the idea of basing depreciation allowance on observed market value of the "rm.
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equilibrium framework and focus on the response of the "rm to the taxation scheme, as in a game between the taxation authority striving for neutrality with respect to extraction path, and the "rm, striving for maximum after-tax pro"t. The interest rate is taken as given, as would be the case for a small open economy, or for an economy with constant marginal product of capital over a relevant range. Depreciation allowances are a central topic in mineral economics (Heaps and Helliwell, 1985), and we are dealing with the theoretically ideal depletion allowance for extractive "rms with non-durable outputs (as in oil, coal, etc., as distinct from copper, gold, etc.). For an extractive "rm, the decision at each date is one of optimal disinvestment from its current stock of reserves. Though formally optimal investment and disinvestment can be analyzed the same way under Samuelson taxation, the latter turns out to be simpler because the "rm has no incentive to borrow and its single-state variable (its remaining stock) moves in a simple fashion with respect to its single control variable (its current extraction). In an optimal production}investment problem, the possibility of borrowing makes the link between the current production decision (the control), and the state variable (the capital stock of the "rm) complicated because current investment is another control variable for the "rm. Hence our analysis, based on the purely extractive "rm, is in a sense prior to that of the producing-investing "rm. We show that a Markovian depreciation rule would provide the incentive for a "rm not to deviate from its best extraction (disinvestment) path for the no-tax scenario and has the subgame perfectness property: even if a "rm has deviated (say accidentally) at some time, it will not have an incentive to deviate in the future from its best path achievable given the current state. Our invariance procedure can be described as follows: have the tax authority work out the "rm's optimal extraction path under the no-tax scenario and, given this path, have the authority obtain the time path of true economic depreciation, contingent on the state of the "rm. The tax authority then assigns this depreciation allowance to the "rm for its optimization under the Samuelson taxdepreciation allowance scenario. Value of the "rm invariance and extraction As pointed out by a referee, if each level of Samuelson's tax results in a distinct interest rate, then true value invariance will never obtain. But a standard assumption in partial equilibrium analysis is the rate of interest does not change. For example, Bond and Devereux (1995, p. 59) assume that &the current valuation of the entitlement 2is not a!ected by the introduction of the tax. This assumption parallels the requirement under certainty that the nominal discount rate is unchanged by the presence of the tax'. For analysis with extractive "rms, see Dasgupta and Heal (1979), Heaps (1985), and Gaudet and Lasserre (1986). Long and Sinn (1984) focus on extractive "rms and the neutrality question, but in a speci"c general equilibrium framework. The neutral business tax for producing}investing "rms is addressed in Sandmo (1974,1979), Stiglitz (1973, 1976), King (1974, 1975), Harberger (1980), Boadway and Bruce (1984), Fane (1987), Lyon (1990), Bond and Devereux (1995), and others.
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path invariance obtain. The invariance depreciation formula must be tied to the state variable in order to prevent the "rm from reacting to the formula and, roughly speaking, the invariance formula must be derived in the no-tax environment in order that the "rm assigns the correct shadow price to the state variable under the tax regime. We add two new bits to the Samuelson tax invariance literature: (a) we demonstrate that the &true economic depreciation' in the tax depreciation scheme must be derived in the no-tax scenario, and (b) we demonstrate that the depreciation allowance must be &state-based' in order to rule out deviations by the taxed "rm. We also provide a rigorous economic analysis of the concept of a &depletion allowance' for mining economics. Several authors have paid attention to economic depreciation in the context of intertemporal optimization; but they have not dealt with the examination of alternative tax rules that supposedly represent Samuelsonian true economic depreciation, and the possibility of deviation by the "rm. In fact, some authors have chosen to work with a diwerent concept of economic depreciation. Sandmo (1974) de"ned the &true' rates of depreciation as the true rate of physical wear-and-tear rate plus the interest rate, minus the rate of increase in the price of the investment good. Atkinson and Stiglitz (1980, p. 142) seemed to equate Samuelsonian true economic depreciation with the replacement cost of physical wear-and-tear. Howitt and Sinn (1989), by abstracting from natural resources and pure pro"t, dealt with a model with much simpli"ed economic depreciation, as invariance was not a central concern of their paper. None of these papers contemplates the possibility that the "rm may deviate from a path in order to reduce its tax burden. In contrast, we focus precisely on the issue of the "rm trying to reduce its tax burden, given a depreciation allowance rule. We emphasize the possibility that a tax rule which would ensure value invariance if the "rm would commit to a given path may not work if the xrm can deviate. In this respect, our paper is in the same spirit as that of Karp and Livernois (1992), but that paper did not deal with Samuelsonian economic depreciation: the authors were addressing the question of regulating a monopolist so as to achieve the extraction path that a social planner would choose. They reported on alternative taxing procedures, in the case of monopoly, which would lighten the information required by the tax setting authorities. Implementing the Samuelson tax scheme is information intensive. The "rm must "le its cost structure with the tax authorities at time zero. Contingent on its stock, s(t), of remaining reserves, the government calculates a depreciation allowance rate (s(t)) which will result in the "rm extracting the optimal quantity q(t). The rate (s(t)) will change in general as the stock size shrinks. Samuelson (1964) was not explicit about how this tax scheme was to be implemented but the implicit rate in the depreciation allowance would also vary in his model. The analysis in Samuelson (1964) is very short and leaves one with the impression that value invariance and its associated path invariance is easy to implement. Clearly, neutrality is an obvious desideratum of a tax scheme. Our analysis
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suggests that neutrality is not easy to be achieved via legislated &true economic depreciation'. Like many results in optimal taxation, Samuelsonian invariance (value and path) is hard to implement because the tax authority must "ne tune its &rules' for each "rm. The rest of this paper is organized as followed. Section 2 investigates, in a context of a very simple model of a resource extracting "rm, the incentive for the "rm to respond to the various depreciation allowance rules by deviating from its best production}investment path in the no-tax scenario. The correct economic depreciation rule is then derived, and is shown to have the Markovian property. Section 3 generalizes the result. Section 4 o!ers several additional examples.
2. A simple model Consider a resource extracting "rm. The initial stock in the ground is s . The rate of extraction at time t is q(t). Let s(t) be the stock available at time t. Then ds s (t), "!q(t). dt
(1)
The cost of extraction is C(q), where C(0)"0, C'0 and C'0. To simplify, assume that the "rm sells its output in the world market, at an exogenously given price p which is constant over time. The rate of interest is r, a positive constant. It is assumed that r does not change with the tax rate. This is the case if the country is a small open economy facing a given world interest rate. In the absence of tax, the "rm wishes to "nd an extraction path q(t)50 and a terminal date ¹ to maximize the integral of discounted cash #ow:
2 e\PR[pq(t)!C(q(t))] dt (2) subject to (1), and s(0)"s , s(¹)"0. It is well known that the optimal path must satisfy the following Euler equation, called the Hotelling rule, which says that the net price (i.e. price minus marginal cost) must rise exponentially at the rate of interest: 1 d(p!C) "r. p!C dt Since p is constant, this rule reduces to !C(q(t))q (t)"r[p!C(q(t))].
(3)
This di!erential equation and the following conditions: q(¹)"0
(4)
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and
2 q(t) dt"s (5) uniquely determine the optimal time path qH(t) and the optimal exhaustion time ¹H. If we substitute qH(t) and ¹H into integral (2), we obtain the value of the "rm at time t"0, which we denote as JH(0). Similarly, the value of the "rm at time t50 is de"ned as
2H 2H e\PV\R[pqH(x)!C(qH(x))] dx, e\PV\RH(t) dt, (6) R R where H(t) denotes the pro"t at time t. From this, we can compute the rate of change in the value of the "rm JH(t)"
JQ H(t)"rJH(t)!H(t).
(7)
Economic depreciation is de"ned as the negative of this rate of change DH(t),!JQ H(t).
(8)
Now consider the introduction of a tax scheme whereby pro"t, net of some (at present unspeci"ed) depreciation allowance which we denote by D (t), is taxed at O a rate . Interest income is also taxed at this rate, and interest payments are tax deductible. If the xrm does not change its extraction path, then its value at time t in this tax scenario is
2H e\P\OV\R(1!)[pqH(x)!C(qH(x))]#D (x) dx, (9) O R where the subscript in JH(t) indicates that the value is computed in the tax O scenario. Di!erentiate JH(t) with respect to time, we get O JQ H(t)"(1!)rJH(t)!(1!)H(t)!D (t). (10) O O O Clearly, if D (t)"!JQ H(t), then (10) reduces to O O JQ H(t)"rJH(t)!H(t), (11) O O which is identical in form to Eq. (7). This fact, together with JH(¹H)"JH(¹H)"0, implies that JH(t)"JH(t) for all t. And the depreciation O O allowance that permits this invariance is D (t)"!JQ H(t)"DH(t),!JQ H(t), i.e., O O one can use the depreciation in the no-tax scenario as the depreciation in the tax scenario. This is of course just a restatement of Samuelson's invariance result. It serves as a departure point for our main question: would the "rm have an incentive to change its extraction path? JH(t)" O
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Clearly, if the government announces the tax rate and the time path DH(t) of allowance for depreciation, regardless of whether the "rm might change its extraction path, then the "rm would seek to maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#DH(t) dt
(12)
while taking the time path DH(t) as given. The Euler equation for this problem is 1 d(1!)(p!C) "(1!)r, (1!)(p!C) dt which implies !C(q(t))q (t)"(1!)r.
(13)
This equation is di!erent from (3). This shows that the "rm will have an incentive to change its extraction path. The reader would perhaps object to this argument, on the ground that in the objective function (12) we have inserted a depreciation allowance which corresponds to the &best' extraction path (under the no-tax scenario), and not a truly Samuelsonian economic depreciation that corresponds to the "rm's actual extraction path. This point is well taken. We are well aware that in (12), we do not present the Samuelsonian scheme, and that nobody would seriously advocate the depreciation scheme represented in (12). We have chosen to introduce this naive scheme only as a device to start o! some meaningful discussion. The point is, for each feasible time path q(t), one needs a corresponding D (t) so that the Samuelsonian invariance result applies. This O creates a tremendous informational obstacle to the implementation of the Samuelsonian scheme: a "rm must supply to the tax o$ce, at time t, information about its extraction rate for all points of time beyond t. For otherwise one cannot calculate the change in the value of the "rm at time t. And the tax o$ce must verify that such plan is actually followed by the "rm. We refer to this depreciation allowance scheme as B, and the one represented in (12) as A. Scheme B is informationally very expensive. In what follows, we locate a few other schemes that lie between these two extremes. The reader is forewarned that the solution ultimately presented also requires a great deal of information. We have seen that depreciation scheme A does not ensure invariance, because the depreciation allowance DH(t) is independent of the action that the "rm might choose, even though it is derived from the pro"t-maximizing behavior of the "rm in the no-tax scenario. Consider again the resource-extracting "rm in the no-tax scenario. It is well known that the current-value Hamiltonian at time t equals rJH(t). Thus, rJH(t)"HH(t)"(qH(t))! H(t)qH(t),
(14)
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where H(t) is the shadow price of the resource stock, which by the maximum principle, equals the net price H(t)"p!C(qH(t)).
(15)
From (14), (15), (7) and (8), we obtain DH(t)"[p!C(qH(t))]qH(t),
(16)
which says an eminently sensible thing: economic depreciation for the resource extracting "rm is equal to the amount extracted multiplied by the scarcity rent. Note that if the xrm does not change behavior, this depreciation results in no tax on the marginal unit of production: pre-tax cash #ow is p!C(qH(t)) on the last unit of production, which is equal to the marginal depreciation allowance. So it would seem that the following depreciation allowance scheme might ensure Samuelsonian invariance: If the "rm extracts q(t) (which need not be the same as its extraction in the no-tax scenario), the tax authority allows it to deduct the amount [p!C(q(t))]q(t). We call this economic depreciation scheme C. Under this scheme, the "rm would maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#[p!C(q(t))]q(t) dt (17) subject to the usual constraints. However, it is clear that the Euler equation for this problem involves the third derivative C(q). So this scheme would make the "rm choose some extraction path that is di!erent from qH(t), and invariance would not be obtained. To avoid a third derivative in the Euler equation, one might be tempted to use the following modi"ed scheme, called the modixed C economic depreciation scheme: Allow the "rm to deduct [p!C(qH(t))]q(t) where p!C(qH(t)) is evaluated along the optimal path obtained in the no-tax scenario, while q(t) is the actual extraction at t. The "rm would then maximize
2 e\P\OR(1!)[pq(t)!C(q(t))]#[p!C(qH(t))]q(t) dt, (18) where [p!C(qH(t))] is treated as a known function of time, and we denote it as
(t). The "rm takes this function as exogenously given, because its choice of the path q(t) does not a!ect this function. The Hamiltonian is H"(1!)[pq(t)!C(q(t))]# (t)q(t)! (t)q(t). O
As can be veri"ed on the examples in Section 4, our proposed solution has the property that, even if the "rm tries to contemplate deviation so as to reduce the tax burden, the depreciation allowance results in no tax on the marginal unit of production: pre-tax cash #ow on the last unit of production is equal to the marginal depreciation allowance. We thank a referee for pointing this out.
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Unfortunately, this yields an Euler equation which is quite di!erent from the one in the no-tax scenario. We move on to another formulation. We are now introducing a scheme that does ensure the Samuelsonian invariance. Let <(s) be the value function of the mining "rm's optimization problem under the no-tax case. More precisely, let
2 <(s)"max e\PV\R[pq(x)!C(q(x))] dx O 2 R subject to s(t)"s,
s "!q
and s(¹)"0. We know that <(s(t))" (t)"p!C(q (t)) for any s(t)50 where q (t) is the optimal extraction rate at time t given that the "rm optimizes from time t with the starting stock s(t), in the no-tax scenario, i.e., q "q(s) (where the super script opt indicates optimal choice, the subscript n refers to the no-tax scenario, and the subscript refers to the situation with Samuelsonian taxation). Let us try the following depreciation allowance formula, which we call the stock-andyow-dependent depreciation scheme, or the E scheme. d<(s(t)) D (t, s(t), q(t))"! "!<(s(t))s (t)"<(s(t))q(t). O dt
(19)
Then, under the tax scenario, the "rm maximizes
2 e\P\OR(1!)[pq(t)!C(q(t))]#<(s(t))q(t) dt (20) subject to the usual constraints. (Note that this is a problem for the extractive "rm under the tax scenario, and <(s(t)) has been obtained for this "rm under the no-tax scenario.) The Hamiltonian for problem (20) is H"(1!)[pq(t)!C(q(t))]#<(s(t))q(t)! (t)q(t) O and we get the necessary conditions H "(1!)(p!C) #<(s)! "0, O O q
(21)
(22)
Q "r(1!) !<(s)q. O O Di!erentiate (22) with respect to t to get
(23)
(1!)C(q)q "<(s)s ! Q . O
(24)
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From (23) and (24), !C(q)q "r , O which can be expressed as
(25)
d [p!C] "r O O dt or as !C(q)q "r[(1!)(p!C)#<(s)].
(26)
But since <(s)"p!C(q(s)), where q(s) is optimal under the no-tax scenario, Eq. (26) becomes !C(q)q "r(1!)(p!C(q)) #(p!C(q)). O This reduces to the Euler equation (or r% rule) for the no-tax extraction problem when C(q)"C(q)). We now provide a formal demonstration that the two extraction paths q(t) and q(s(t)) will be the same, given <(s)"p!C(q(s)), i.e., under the depreciation scheme D (t)" O [p!C(q(s))]q(t). This proof relies on the Hamilton}Jacobi}Bellman (HJB) equation. For the no-tax case, the HJB equation for the resource extracting "rm is r<(s)"max [pq!C(q)!<(s)q]. (27) O Assume that we have found a function <(s) that satis"es the functional equation (27). From this, we can obtain the function <(s)q (for the no-tax case), which we now use as the depreciation allowance in the tax scenario. Then the value function in the tax scenario, denoted by < (s), must satisfy the following O equation: (1!)r< (s)"max [(1!)(pq!C(q))#<(s)q!< (s)q]. O O O
(28)
Clearly this functional equation has as solution < (.)"<(.), because if we O substitute this proposed solution into (28), we will get (1!)r<(s)"max [(1!)(pq!C(q))#(1!)<(s)q] (29) O and <(s) clearly satis"es this functional equation, because it satis"es the HJB equation (27) which is identical to it (since the terms (1!) on both sides cancel each other out). This proof makes clear that the depreciation formula for invariance (or neutrality) must be based on the true economic depreciation for the "rm's stock
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under the no-tax scenario, not true economic depreciation for the "rm's stock under the tax scenario, even though, ex-post, the two are identical. A key insight in this regard is: the shadow price of a unit of stock moves at r% under the no-tax scenario, and at (1!)r% (minus an adjustment term, see (23)) under the tax scenario. The shadow prices are distinct in general but will be the same under the invariance tax scheme. The general result could be stated as follows: the neutral tax scheme results in < (s), the shadow price for stock under the tax O scenario, made the same as <(s), the shadow price for stock under the no-tax scenario. This completes the proof that our stock-and-yow-dependent depreciation formula D"<(s)q ensures the Samuelsonian invariance, even though the modixed C economic depreciation formula, which is very similar to it, does not. Even though <(s)"p!C(q) when q is optimally chosen given s, i.e. when q"q(s), it is incorrect to use [p!C(q)]q as the depreciation formula, because this would not re#ect the fact that the scarcity value depends on the stock level s. (It would be correct, however, to use [p!C(q(s))]q.) The following proposition sum marizes our main result so far: Proposition 2.1. For the resource extraction xrm in our model, the correct depreciation formula that ensures the Samuelsonian invariance is D"<(s)q, or equivalently, D"[p!C(q(s))]q, but not D"[p!C(q)]q. Our result indicates that for the real-world application of the Samuelson tax scheme, quite a lot of information is needed. The tax authority must collect data on each "rm on costs and stock characteristics, and solve the optimal extraction program, under the no-tax scenario, and then work out true economic depreciation, with the time path of the shadow price of the stock expressed as function of the remaining stock. The depreciation allowance, q(t)<(s(t)) is then assigned to the extractive "rm at each date. 3. Generalizations We now generalize our formula to allow for (i) the possible time-dependence of the cost and revenue functions, and (ii) possible natural growth of the resource. We replace the expression pq!C(q) by the more general net cash #ow function N(s, q, t), and we assume that s "f (s, q, t). In the no-tax scenario, the Hamilton}Jacobi}Bellman (HJB) equation for the "rm's optimization problem is
r<(s, t)"max N(s, q, t)# O
<(s, t) <(s, t) f (s, q, t)# s t
.
(30)
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We assume that this functional equation has a solution, which we denote by <(s, t), which is the value of the "rm at time t if the stock at that time is s. We de"ne the economic depreciation as
d<(s, t) <(s, t) D(s, q, t)"! "! dt s
f (s, q, t)!
<(s, t) . t
(31)
Now consider the scenario with a tax tate and some depreciation allowance A(s, q, t). The HJB equation for the "rm is then (1!)r< (s, t) O
< < O f (s, q, t)# O "max (1!)N(s, q, t)#A(s, q, t)# s t O
, (32)
where < (s, t) is the value of the "rm at time t, in the tax scenario, if the stock at O that time is s. What form should A(s, q, t) take to ensure that < (s, t) is the same as <(s, t)? O Proposition 3.1. If the government sets the depreciation allowance rule to be the same as the function describing true economic depreciation in the no-tax scenario, i.e., if it sets
A(s, q, t)"!
<(s, t) <(s, t) f (s, q, t)! , s t
(33)
then Eq. (32) has the solution < (s, t)"<(s, t), i.e., the value of the xrm in the tax O scenario is equal to its value in the no-tax scenario. Proof. Substituting (33) into (32), we see that the resulting functional equation has <(s, t) as solution. (In other words, replacing < (s, t) (and its partial derivaO tives) by <(s, t) (and its partial derivatives) in that functional equation, we obtain an equation that is identical to (30), except for the terms (1!) on both sides, which cancel each other out.) Remark. Proposition 3.1 is also correct if r and are functions of time. It is easy to see that our proof can be modi"ed to accommodate this time-dependence of r and . The case where s and q are vectors can also be easily accommodated. Strictly speaking, the transversality condition lim e\PR<(s, t)"0 must be satis"ed. R
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4. Some examples In this section, we consider some examples that illustrate how the Samuelsonian true economic depreciation should be represented to ensure that the "rm does not deviate from its best path in the no-tax scenario. 4.1. Example 1 Consider a monopolist that costlessly extracts a natural resource. Let q(t) denote the rate of extraction, and s(t) the resource stock. Then s (t)"!q(t).
(34)
The demand function is p"q\?, 0((1.
(35)
The rate of interest is r, a positive constant. In the absence of taxation, the "rm maximizes the present value of the net cash #ow N(t)"p(t)q(t)"q(t)\?, subject to q50 and lim s(t)"0 R
(36)
s(0)"s (given). This problem yields the necessary condition q "!rq, hence q(x)"q(t)e\PV\R ?.
(37)
(38)
Eqs. (34)}(38), imply rs(0) q(0)"
(39)
and hence the optimal extraction rule is rs(t) q(t)" .
(40)
The value of the "rm at time t is therefore
? q(x)\?e\PV\R dx" s(t)\?. r R It is easy to verify that this solution satis"es the HJB equation <(s(t))"
r<(s)"max[q\?!<(s)q], where the maximization is with respect to q50.
(41)
(42)
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Now introduce an income tax at the constant rate . The "rm's after tax net cash #ow at time t is (1!)q(t)\?#D(t),
(43)
where D(t) is the Samuelsonian true economic depreciation:
D(t)"!
? s(t)\?q(t). r
(44)
This depreciation formula involves both stock and #ow, and implicitly takes into account the pure pro"t, which is re#ected in the elasticity of demand parameter in (44). When there is tax on capital income and where economic depreciation is tax deductible, the HJB equation is
? (1!)r< (s)"max (1!)q\?#(1!) s\?q!< (s)q . O O r
(45)
One can verify that the following value function satis"es (3.12):
? < (s)" s\?. O r
(46)
Comparing (46) with (41), we con"rm that the value of the "rm is independent of the tax rate, and the tax has no e!ect on the "rm's optimal extraction path. To conclude this example, we note that the above invariance result can also be veri"ed by using the standard methods of optimal control theory. One would then obtain the following Hamiltonian to be maximized with respect to q
H"(1!)q\?#(1!) with
? s\?q! q r
(47)
Q "(1!)r #(1!)
? s\?\q. r
(48)
The following necessary condition can then be derived after some manipulation:
!(1!)q\?\q "r(1!)q\?#r(1!)
? s\?. r
(49)
Again, setting q"rs/ (and therefore q "rs /"!rs/), we "nd that (49) is indeed satis"ed. 4.2. Example 2 A competitive "rm extracts a natural resource at the cost c per unit. Its net cash #ow at t is N(t)"[p(t)!c]q(t), where the "rm takes p(t) as given. In the
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industry equilibrium with n "rms having identical unit cost c, the price path must obey the Hotelling rule p "r(p!c).
(50)
It is clear that the value of the "rm at time t, when its resource stock is s(t), is <(s(t), t)"(p(t)!c)s(t).
(51)
Indeed, (51) satis"es the HJB equation r<(s, t)"max [(p(t)!c)q!< q#< ], Q R O
(52)
where from (51) < "p(t)!c, < "p (t)s(t)"r(p!c)s(t), where the last equality Q R follows from (50). The "rm's rate of extraction is of course indeterminate because of the constant unit cost assumption. The "rm's economic depreciation is D(t)"!
< < O q# O "max (1!)(p!c)q#(p!c)q!p s! O s t
.
Again, < (s, t)"<(s, t)"(p(t)!c)s solves this functional equation. Value inO variance is again con"rmed.
5. Conclusion We have shown that when the application Samuelson's value invariance theorem to the case where "rms optimize intertemporally requires a judicious representation of economic depreciation so that the "rm would not deviate from the production}investment path that it would choose in the no-tax scenario. While a given path of economic depreciation can be represented in several di!erent ways, only the Markovian representation does not distort incentives. This representation ensures &subgame perfectness'. In this paper, we have restricted attention to the polar cases where the "rm is either a monopolist or a perfectly competitive "rm. An extension to the case of
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oligopoly would not be trivial, because under oligopoly, each "rm's value depends not only on its stock level, but also on the stock levels of its rivals. In that case, a "rm's true economic depreciation would seem to depend on its rivals' actions. This is a challenging research topic that is part of our future research agenda.
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