Leviathan governments and carbon taxes: Costs and potential benefits

EUEVIER

European Economic

Review 39 (1995) 1215-1236

Leviathan governments and carbon taxes: Costs and potential benefits Franz Wirl a,*, Engelbert Dockner b aInstitut fir Energiewirtschaft,

Technische Unioersitiit Wien, Guphausstrasse Austria b Uniuersitiit Wien, Vienna, Austria

Received December

27-29, A-1040 Wien,

1993; final version received August 1994

Abstract This paper addresses positive aspects of the global warming debate that so far has been concerned by and large with normative issues. The paper considers a consumers’ government that appreciates tax revenues as such (‘Leviathan’) in addition to conventional measures of welfare; i.e., carbon taxes serve the dual purpose of correcting for externalities - here global warming, or more generally, a stock externality from energy use - and of raising revenues. This government faces either a competitive or a perfectly cartelized supply. The major findings are: (i) consumers may benefit from a Leviathan government that appropriates some of the monopoly rent by deterring preemption to some extent; (ii) the Leviathan motive raises initial taxes and thereby lowers initial emissions but increases the long-run stock externality (in a play with linear strategies); (iii) there exists a continuum of equilibria in nonlinear Markov strategies, which are, however, Pareto-inefficient compared with the linear strategies; (iv) but in case of a binding resource constraint, nonlinear strategies are the only feasible ones. Keywords:

Leviathan

JEL classification:

* Corresponding

governments;

Carbon taxes

Q30, H20, C73, D78

author

0014-2921/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO14-2921(94)00036-O

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Economic Review 39 (1995) 1215-1236

1. Introduction The issue of global warming receives considerable attention in the recent scientific literature. The economics profession considered so far by and large the normative case of efficient strategies to mitigate the potential threats from global warming; see e.g., Nordhaus (1990, Manne and Richels (1991a, b), Pearce (1991), Morgenstern (19911, Schelling (1992), Dombusch and Poterba (19921, Peck and Teisenberg (1992). Hoe1 (1992, 1993) and Tahvonen (1993) are recent papers that consider particular strategic aspects. For an account of the scientific basis of the greenhouse effect see Schneider (1989), Cline (1991) and the comprehensive report by the U.S. Office of Technology Assessment, U.S. Congress (1991). Non-competitive and strategic energy producers may preempt carbon taxes by raising the price at front if the consumers’ government taxes in order to internalize external costs, see Wirl (1994). This strategic response of suppliers may induce a considerable transfer of wealth such that the costs of carbon taxes borne by consumers may be underestimated in the above quoted approaches. Indeed, Saudi Arabia as the leading OPEC member proved prior to the Earth Summit in Rio de Janeiro, June 3-14, that it can preempt an EEC carbon tax by raising the price at front. According to The Economist (19921, the oil price increase of almost $4 per barrel Arab Light between March and June 1992 was due to strategic OPEC reactions ‘to see the price rise by $3 a barrel to match the effect of the first step in the EC’s new carbon-tax plan’. And it appears that OPEC succeeded at that time as the EEC withdrew its proposal. However, the scope for such a preemption by producers would be considerably weakened if the government were not entirely benevolent but used green arguments as a figleave to raise tax revenues. This conjecture prompts this investigation. More precisely, the government is not benevolent but instead and at least to some extent (but not entirely) a Leviathan in the sense of Brennan and Buchanan (1980). Indeed, this revenue motive behind energy taxation seems more or less obvious although politicians prefer to camouflage it as an environmental objective: in the U.S., all the various proposals to raise the tax on petrol are intended to increase government’s revenues; in the FRG, the increase of petrol taxes since 1991 serves deliberately as a source for the financial transfer from the West to the East. In fact, positive aspects of environmental policy making are rarely treated in the literature that employs by and large the normative (and therefore sometimes naive) approach, see e.g., the text books of Baumol and Oates (1988) and Pearce and Turner (1990) and the survey, Cropper and Oates (1992). A possible advantage of a Leviathan government is that it limits the scope for suppliers’ preemptions. And similarly, many Greens do not care about underlying Leviathan motives if these taxes help the green issues. Given these motivations and put shortly, this paper investigates carbon taxes set by a government, which is a blend of the neoclassical benevolent despot and public choice’s Leviathan.

F. Wirl, E. Dockner / European Economic Review 39 (1995) 1215-1236

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Section 2 introduces the model. In Section 3 we investigate competitive and in Section 4 we derive Markov perfect Nash equilibria (linear and nonlinear) between an energy pricing supply cartel and a taxing consumers’ government. Lengthy and cumbersome derivations and proofs are relegated to an appendix. Concluding remarks complete this investigation.

2. The model The model intends to capture the central, i.e., the dynamic, economic and strategic features, of the global warming debate. However, no attempt will be made to model accurately the so far acquired scientific knowledge of the greenhouse effect, for research in this direction see Nordhaus (1992). In fact, the model considers a simple stock externality of which carbon dioxide is just the most discussed example. Nevertheless, the analysis continues to use CO, as pars pro toto for stock externalities but omits the scientific details of the accumulation of carbon dioxide in the atmosphere. Following Wirl (1994) we consider a linear and time invariant energy demand function f(?r)=a+br,

a>o,

b
(1)

that depends on consumer price 7r; the root of the demand function f determines the choke price rr’: = -u/b. The consumer price is the sum of the wellhead price p and the tax r: rr=p+r.

(2)

The government u(r):=

accounts

(“&)dn=

for the consumers’ -$a*/b-

benefits:

conventional

surplus

U,

(m++br2),

the costs associated with the stock externality X (e.g., the costs associated with a rise global temperature due to increased CO, concentrations in the atmosphere),

C(X) =

;cx*,

(4)

and the transfers from tax receipts (rf). These tax revenues are returned to the consumers so that taxation involves only a triangle loss of consumers’ surplus; this assumption is common, see e.g., Nordhaus (1993). Although, these tax revenues will be reimbursed to the consumers, politicians appreciate these tax revenues (of) as such, because the tax revenues measure the power of the politicians: higher tax revenues increase the discretion of the politicians about what and whom (including themselves) to subsidize; for further motivations of this Leviathan motive of

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Economic Review 39 (1995) 1215-1236

governments see the public choice literature, in particular Brennan and Buchanan (1980) and Mueller (1989). The weight parameter A 2 1 accounts for this Leviathan motive and the limiting case A = 1 corresponds to a benevolent government. Summarizing, the government is supposed to tax energy in order to maximize the following objective:

+AbWf( p(t) + W>l) dt.

(5)

A higher parameter value of A means that the rulers care more about the tax revenues and less about conventional consumers’ welfare, i.e., A measures the importance of the Leviathan component in the objective (5); future benefits are discounted by the factor r. Put shortly, this government is partially a Leviathan and partially benevolent or put more modest, it is rewarded (e.g., at elections) for delivering consumer surplus. The evolution of cumulative energy consumption determines simultaneously the CO, concentration in the atmosphere:

it(t) =f(p(t)

+7(t)),

X(0)

=O, X(t)


(6)

which provides the dynamic constraint. In short, energy consumption involves a stock externality. This equation simplifies by neglecting stock depreciation, which is very sluggish (around 200 years) and nonlinear, and other non-energy related greenhouse gases; Hoe1 (1992) uses the same simplification. On the other hand, the identity between energy consumption and CO, released into the atmosphere (roughly, just half of the carbon emissions enter the atmosphere) is not crucial as we can measure energy in that unit that releases one ton of carbon to the atmosphere. Furthermore, we assume that cumulative energy extraction is not constrained by the resources in the ground, denoted R, but by its impact on the climate, i.e., the inequality constraint in (6) is not binding when carbon taxes are imposed. This assumption stipulates that global warming constrains energy use no matter what are the precise parameter values of: demand, external costs, the Leviathan’s appetite and the discount rate. This assumption is quite plausible considering the vast amounts of cheap coal and simplifies the analysis because linear, feedback strategies determine the outcome. However, we will sketch the (nonlinear) solution when the resource constraint is binding, see Section 4.3. Complementary to the theoretical analysis, we introduce an example in order to highlight numerically and graphically the general and theoretical propositions and to verify the claim that a Leviathan government may be better, not only compared with muddling through and no intervention, but even when compared with a benevolent government. However, this example does not pretend to give an

F. Wit-l, E. Dockner / European Economic Review 39 (1995) 1215-1236 Table 1 Numerical

1219

example

Demand Discount rate r External costs c

f(n) = 12.72-0.028~ 0.03 0.034

+ rrc = 454.3

empirically relevant analysis of carbon taxes and for this reason, a discussion the parameters in Table 1 will be suppressed. ’

3. Competitive

of

supply

From the assumption that cumulative consumption is not bound by the resources in the ground follows that a competitive industry supplies the considered fossil energy at the marginal cost. For the sake of simplicity, these marginal costs are constant and negligible. Therefore, competitive supply ensures p(t) = 0 and the government’s strategy boils down to maximize (5) subject to (l)-(4) and (6). This is a standard optimal control problem. Proposition 1 characterizes the optimal strategy rrc = 7’; the superscript C (capital letters to differentiate from ’ used to identify the choke price) refers to the competitive supply; the subscript a identifies the stationary solution, e.g., X, denotes the ultimate stock of pollution. Proposition 1. The Leviathan’s optimal tax policy can be written as a linear feedback law and it is given by the following formulas: 7=(x)=7rc+(+c(x-x~)=7rC(x), aC=i[r-dm]/b, X,” =A[ -ar/(

(7a) 0=1/(2A-1),

bc)].

(7b) (7c)

Proof: Appendix A derives (7) and all the following economic properties are a direct consequence of the explicit formulas (7). However, the stationary solution can be derived directly using economic arguments without lengthy calculations. From the definition of a steady state follows that the demand must vanish, thus

’ Nevertheless, for those who are interested how the numbers are made up, a brief explanation follows: a linear demand function is calibrated for present world energy demand (roughly eight billion tons of oil equivalent) and an aggregate primary energy price (roughly $100 per ton of oil equivalent) assuming a price elasticity of - 0.2. However, the accounting is done in terms of tons of CO, taken up by the atmosphere: one ton of fossil fuels releases approximately 2.35 tons of CO, (based on OTA (1991) and the U.S. energy pattern) of which, according to Nordhaus (19911, only half enters the atmosphere. Thus, dividing the reported taxes roughly by two gives the tax per ton CO, emitted. The cost parameter c is set such that it is socially efficient to add only up to 400 billion tons of CO, to the present concentration in the atmosphere. Therefore, large amounts of coal will remain unused, at least for a benevolent government.

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Economic Review 39 (1995) 1215-1236

Emissions over time

Strategies (state space)

-I 0

100

400

wo

000

0

1. Comparison

40

80 the

X

Fig.

20

of tax strategies and the associated flow of emissions Leviathan governments; parameters from Table 1.

80

$00

(0

of benevolent

versus

7, = rrc. Moreover, the benefit from emitting this last ton of carbon (and consuming the corresponding amount of fossil fuels) must equal the present value of increasing for all future the external costs by this marginal unit. The Leviathan’s benefit from allowing the emission of this last ton is equal AT, =AT’ = -As/b (the consumer’s surplus vanishes). Equating this benefit to the present value of marginal external costs

AT, = -Au/b

= /-,C”c’(

X,)dt

0

= cX,/r,

(8)

and solving (8) for X, yields (7~). The explicit formulas (71, which include the benevolent government for A = 1, state that a Leviathan government chooses too high taxes at the beginning, t = X = 0, becauserc(0) increases for a larger parameter value A, but too low taxes later so that cumulative emissions increase and thereby long run environmental damages. In fact, stationary pollution is inflated exactly by the factor A,X: = AX,* where the asterisk refers to the socially eficient solution (A = 1). Fig. 1 highlights the qualitative properties of the linear strategies (7) - the tax increases up to the choke price, consumption (= emission) declines to zero - and the differences between a benevolent and a (moderate) Leviathan government. The left-hand side shows the tax strategies in the state space while the right-hand side compares the flow of emissions over time. The right-hand side documents that the period where the Leviathan is the conservationist’s friend of may be rather short; however, this

F. Wirl, E. Dockner/European

Economic Review 39 (1995) 1215-1236 benevolent

Fig. 2. Impact of the Leviathan

1221

Leviathan

motive on the tax strategy.

figure underestimates slightly the ‘green’ period of the Leviathan because the stock matters and that is lowered over more than thirty years. What is the reason that the Leviathan motive leads to more pollution despite raising taxes for some time? For this purpose consider a pure Leviathan (A very large), who would keep the tax at the revenue maximizing level, i.e., 7(t) = i 7~’ for all f, resource constraints aside. Hence, any inclusion of Leviathan motives, even of moderate ones (A > 1 but not too large), tends to center the tax around $rc, which flattens the (linear) strategy compared with the efficient solution. This twist of the tax strategy due to the, possibly moderate, Leviathan motive must lead to a higher stationary pollution, see Fig. 2. Therefore, the Leviathan’s choices of carbon taxes are higher for low values of the state variable and lower for high values of X. This reiterates Laffer’s insight, albeit in a different context, that too high taxes (which would be compatible with the environmental aspects) lower the tax revenues.

4. Monopolistic

supply (Markov-perfect

Nash equilibria)

The objective of cartelized producers is to find a price strategy that maximizes the present value of profits (using the same discount rate r as the government) assuming that the stock externality as such is largely irrelevant to their welfare:

We analyze Markov strategies because these strategies capture essential strategic interactions, provide a subgame perfect equilibrium and are analytically tractable. These (stationary) Markov strategies are characterized as a map f from the state into the control space so that T(t) = T,(X(t)) and p(t) = T,(X(t)) for all t. Defining the optimal value functions VCX,) for the energy supply cartel and W(X,) for the Leviathan government - these functions V and W determine the

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Economic Review 39 (1995) 12151236

maximum of the corresponding objectives depending on the initial conditions, V(X,,) = J * and W(X,,) = .I1 both for X(0) = X0 - the Markov strategies have to satisfy the Hamilton-Jacobi-Bellman equations: ~V(X)=m$$(~+r)+V’(X)f(p+r)],

(10)

rW(X)=mTa{u(p+~)-C(X)+Arf(p+r)+W’(X)f(p+r)}. (11) Carrying

out the maximization

p”=

-$(v’+a/b+

TV=

-W’(X)

on the right-hand

sides of (10) and (11) yields:

T), -(A-l)[r+f/fl]

(12) .

(13)

Pigouvian& Leviathan component The superscript M refers to the Markov-perfect Nash equilibrium for monopolistic energy supply when A > 1. The price in (12) exceeds the static profit maximizing strategy (given by $(z-” - 7)). Therefore, the producers preempt to some extent even a Leviathan’s taxation policy (compared with a myopic monopoly subject to taxation). The tax 7M in (131, which exhibits the same structure as T’, consist of two terms: the Pigouvian element * (-W’(X)) accounts for the loss in the objective due to an increase in X; the Leviathan component, the marginal revenues from taxation is multiplied by the factor [-(A - l)]. Of course, the Leviathan factor vanishes for A = 1 and (13) reduces to Pigou’s suggestion to set the tax equal to the marginal, intertemporal loss of welfare, T = - W’. The effect of the Leviathan component on the level of taxes may be positive or negative such that the Leviathan motive may not only raise taxes but sometimes lowers taxes. Finally, note that the tax is always positive, no matter how high the producers’ price is so that perfect internalization at the wellhead is impossible. This differs from the usual proposition that monopolies may in principle do more than internalize external costs, i.e., the Pigouvian instrument should be a subsidy. 3 The reason is that the world is separated into producers and consumers and the latter do not care about the producers’ surplus. The simultaneous, linear equations (12) and (13) have the following solution for the instruments depending on the derivatives of the (so far unknown) value functions: p”=[b(V’+W’)-A(a+2bV’)]/(3Ab-b), TM =

[(I

-A)(u

- bV’) -2bW’)]/(3Ab

(14) - b).

(15)

2

But not entirely Pigouvian because W’ is the derivative of a Leviathan objective. 3 On this argument see Buchanan (1969); Ebert (1991) and Katsoulacos and Xepapades recent investigations how market structures affect Pigouvian taxes.

(1993) are

F. Wirl, E. Dockner/European Economic Review 39 (1995) 1215-1236

1223

Substitution of this solution (14) and (15) into the functional equations (10) and (11) eliminates the maximization and leads, after some tedious calculations and simplifications, to a pair of quite amenable nonlinear differential equations: rV=-(b(AV’+W’)-aA)2/[b(3A-1)2], rW=

(16a)

2A - 1

-+cX*-

2 [b(AV’+ 2b(3A

Observe, that both value functions consumer price: 7M = [(2A - l)+-

W’) -aA]*.

- 1) in (16) depend on (AV’ + W’) and so does the

(AV’+

W’)]/(3A

- 1).

This regular occurrence of the term (AV’ + W’) facilitates this asymmetric game, including the nonlinear solutions.

(17) a complete

analysis of

4.1. Linear strategies Proposition 2. The linear Markov-perfect Nash equilibrium strategies between a Leviathan government and a monopolistic supplier are unique, stable and can be explicitly calculated (identified subsequently by the superscript Ml: 79(X)

= 7rc + o”(x-Xr”), (2A - l)aM

( 18a) - 2c/r

( 18b) ( 18c) oM = +[(3A

- 1)/(4A

(Y= (4A - 1)/(3A X,” =A[ -ar/(

- l)] [r - al/b,

- 1)2,

( 18d)

bc)] =X,“.

( 18e)

Proof For the derivation of (18) see Appendix discussed below follow directly from (18).

B and all the economic

properties

The economic implications of (18) are similar to Proposition 1: The price declines monotonically to zero, while the tax and the consumer price increase monotonically up to the choke price. The Leviathan government lowers the initial emissions (at t = X = 0) and the initial price p, but increases the initial tax (such that the consumer price increases) and the long-run environmental damage. In fact, monopolistic supply leads to the same steady state as competitive supply, but

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Economic Reuiew 39 (1995) 1215-1236

Consumer prices (state space)

PO0

,,’ ,,/j ,/’

L P

Emissions over time

/’

,,p

:‘iaxee ,,,’

100.

,,,’

:

i

0’ 0

1 200

400

100

800

I

0' 0

20

X Fig. 3. Comparison supply is cartelized;

of a benevolent (A = 1) and a particular parameters are according to Table 1.

40

00

(10

100

tima(t) Leviathan

government

(A = 2), when

protects the enuironment transiently by reducing initial emissions and by delaying the stock externality, i.e., XC(t) > X”(t) f or all t. Long-run pollution is identical to the competitive case because the producers are ultimately willing to supply at their reservation price, p = 0. Hence, the government’s ultimate trade off between tax revenues and external costs is identical to the one outlined in Section 3. The proof that a monopoly slows global warming follows from the fact that ( - l/( ba >) determines the time constant of the linear differential equation X and uM < uc. Fig. 3 compares the final consumer prices, the taxes (and implicitly the producers’ price p) for a benevolent (A = 1) and for a particular Leviathan government (A = 2) over the state space (lhs) and the flow of emissions over time (rhs). This figure summarizes in a graphical and numerical manner the above characterization: the Leviathan raises the initial tax but charges lower taxes later and thereby increases cumulative pollution. As conjectured, the Leviathan lowers the initial producers’ price p quite significantly while the quantitative impact on the consumers’ prices is small. This documents the rent transfer that results from Leviathan governments. Hence, the transient impact of the Leviathan on conservation is negligible (in contrast to the competitive case in Section 3) and the government’s camouflage - high taxes masked as green policies - is considerably shorter compared with the case of competitive supply. Fig. 4 shows how far the Leviathan’s appetite for tax revenues, measured by the parameter A, affects welfare and thereby documents the conjecture expressed in the introduction that a Leviathan may be beneficial for consumers’ welfare. The

F. Wirl, E. Dockner / European Economic Review 39 (19951 1215-1236 ulooo)

OfJ2)

1225

21000

Environmental ._.

Cos~g..” .’.’ .,,..“.

Cormumers’Welfar6 20600

1

1.6

2

2.2

-20000 2

A

Fig. 4. Comparison of net present values of the consumers’ welfare (consumers’ surplus, plus tax revenues minus environmental costs), the producers’ profits and the environmental costs versus a family of Leviathan governments (identified by A); parameters from Table 1.

Leviathan motive increases the present value of the ‘true’ consumers’ welfare (using A = 1 in J’ for the strategies (18) and A > 1) that reaches the maximum at A = 2.2. 4 Of course the environmental costs increase at the same time due to Proposition 2, and so does the conventional consumer surplus (not shown). The energy suppliers’ profit declines first with respect to A as the transfer of rents from producers to consumers dominates, but surprisingly increases for larger values of A. The reason is that the Leviathan motive allows the energy supplier to increase the cumulative sales which compensates (at least in the example in Fig. 4 for A > 2) for the lower prices the energy cartel can charge.

4.2. Nonlinear

strategies

One of the interesting features of continuous time differential games is that linear quadratic games admit nonlinear strategies as equilibria. In this section we are going to determine the nonlinear, differentiable and stable equilibrium strategies. For this purpose, it turns out that a proper sum (as already indicated above), Z: =AV’

+ W’,

(19)

is sufficient to characterize the consumer price (due to (17)), the value functions and ultimately the strategies. Now adding (16b) to A times (16a), using the definition (19) and differentiating yields a nonlinear differential equation for Z:

4 However, this potential gain to consumers may be questioned by rent seeking arguments, of this financial transfer may be socially wasted in a rent seeking contest.

i.e., parts

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Economic Review 39 (199.5) 1215-1236

4

Fig. 5. Phase plane diagram (X, Z) for the differential

equation (20), Z: = AV’ + W’.

It is this particular feature - the Markov strategies correspond to a meta-value function of (AV + W) - that allows to apply techniques similar to Tsutsui and Mino (1990) despite the asymmetry of the game. That is, the above transformation reduces the pair of ordinary, nonlinear differential equations (16) to the single differential equation (201. Proposition 3. Besides the linear Markov strategies, there exist (uncountably many) nonlinear strategies that support a continuum of stocks of pollution befow X,” =X,“, i.e., nonlinear strategies protect (unintentionally) the environment. The corresponding consumer prices (as a function of the state variable) increase monotonically (ultimately with infinite slope) and exceed their linear pendent (with respect to the state). Proof Phase plane technique is applied to analyze the differential equation (20), which consists of a positive coefficient (the first ratio between the squared brackets in (20)) and of a linear fraction. 5 From this second term in (20) follows that Z’ = 0 = > Z = - cX/r and Z’ = &-~0 = > Z =Aa/b = -Arc. In addition, this locus Z = -Arc characterizes the stability condition, more precisely: each stable strategy must end here, which follows formally from solving (17) for that Z that ensures v~” = rr’. These loci, Z’ = 0 and Z’ = f 00, are shown in the relevant fourth quadrant of the (X, Z) plane, Fig. 5. The linear strategy passes through the point where the second ratio in (20) appears indeterminate. Note that Z cannot fall below (--An’), because the corresponding consumer prices exceed the choke price. Imposing stability and continuity on the set of strategies implies that the paths that start above the linear solution cannot be candidates for continuous and stable Markov strategies because they cannot reach (-AT”) smoothly. Therefore,

’ In fact, one can derive an analytical, but implicit, characterization for this class of differential equations. However, this implicit solution Z(X) adds little for our purpose.

F. Wirl, E. Dockner / European Economic Review 39 (1995) 1215-1236

1227

Strategies Mate space)

Strategies (time domain)

X

t

Fig. 6. Example of a particular, stable, nonlinear strategy in the state space and time domain.

solution (bold curve) and comparison

with the linear

nonlinear and simultaneously continuous strategies can only support pollutions below the linear outcome. Indeed, even integral curves that start very close to the linear strategy may lower ultimate pollution tremendously but of course harm the consumers’ surplus; a particular but characteristic example of a stable nonlinear solution of (20) is shown in Fig. 5. The affine transformation of 2 according to (17) determines the consumer prices. Therefore, the nonlinear outcomes for the consumer prices rr appear qualitatively similar to the linear strategies (namely, increasing over state X and time t), although the increases of T tend to infinity (in the state plane) as the stationary level of pollution is approached. Since a stable, nonlinear solution curve Z(X) must stay below the linear one, the corresponding consumer price must be higher (with respect to X). Substitution of Z(X) into (16) gives the value functions V(X) and W(X) and the corresponding derivatives, which in turn determine the strategies. Fig. 6 compares a (particular example of a) nonlinear Markov strategy with the linear outcome. The major difference is that the producers’ price may increase and that this increase towards the end of the game, coupled with the government’s nonlinear tax strategy, leads to a very steep increase in consumer prices (actually, of infinite slope in the state space). An explanation of these simultaneous increases of prices and taxes is that both, government and producers, engage in a contest over rents. The nonlinear strategy follows initially very closely the linear one up to

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Economic Review 39 (1995) 12151236

a point where apparently the producers refuse to agree on further price cuts and instead raise prices. However, the government will not give in and will raise taxes even stronger. Of course, due to the simultaneity of the decisions in a continuous time framework, the causality may work in the other direction with the govemment raising taxes first and the producers retaliating. Either way, the consequence from this contest over rents is that demand may be choked off within a relatively short period of time so that large amounts of resources remain unused (compared with the steady state in linear strategies). These characteristics hold for all nonlinear strategies (except that the producers’ price p may increase from the beginning). Proposition 3 states that nonlinear strategies mitigate (unintentionally) the environmental problems associated with Leviathan governments. Therefore, a play in nonlinear strategies may (further) increase overall and true consumer welfare (not distorted by Leviathan preferences) due to preserving the environment, yet reducing preemption. Already Dockner and Long (1993) show that nonlinear strategies lower externalities but the reason seems quite different. While in symmetric games nonlinear strategies seem to facilitate cooperation and thus support more efficient (long run) outcomes (see the above quoted paper of Dockner and Long (1993) and Feichtinger and Wirl (1993) for another example), the nonlinear strategies in this asymmetric game seem to be on the contrary much more aggressive. This multiplicity of Nash equilibria in Markov strategies games raises the question whether deductive reasoning allows to reduce this set of equilibria (beyond the above already applied restriction to stability and continuity). We provide two arguments in favour of the linear outcome (and thus for a unique solution of the game): efficiency and a formal point (differentiability).

Proposition 4. The smooth and stable but nonlinear strategies are Pareto-inferior to the linear strategies, i.e., both players are better off by restricting themselves to linear strategies. Moreover, only the linear strategies lead to value functions that are differentiable at the stationary pollution.

The intuitive explanation of Proposition 4 is that the linear Markov strategies lead to the competitive steady state, which maximizes (by definition) a weighted sum of tax revenues, consumers’ and producers’ surplus, i.e., an index of joint surplus (Leviathan’s, consumers’ and producers’). This suggests that both players loose from a play in nonlinear strategies because too low levels of cumulative energy consumption harm profits, tax revenues and consumer surplus. Indeed, this is true with the consequence that (government) policies should follow simple, i.e., linear, feedback rules. Of course the Pareto-efficiency addressed in Proposition 4 does not imply that joint benefits are maximized, because the competitive solution of Section 3 maximizes joint benefits. In this sense the Nash equilibrium even in linear strategies entails, as usual, an inefficiency with respect to cooperation.

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F. Wirl, E. Dockner / European Economic Review 39 (1995) 1215-1236

Proof

The payoff of each player can be derived from (16) by setting V(0) = -( bZ, - uA)‘/[

rb(3A

W(0) = -(2A

- ~A)~,‘[2rb(3A

- l)(bZ,

- 1)2],

(21) - 1)2].

Therefore, the payoffs depend on the initial condition and (22) yields:

av(o)/az, = -2( bz,, - uA)/[

t = 0 =X:

r(3~

(24

Z,. Differentiation

of (21)

- I)*],

(23)

aw(o)/az, = -(2A - l)( bZ, - uA),‘[ r(3A - :I,‘].

(24)

These derivatives (23) and (24) must be positive, because economically feasible initial conditions of Z, must satisfy Z, > -Arc so that (bZ, - aA) < 0. Therefore, aV(O)/dZ, > 0 and aW(O)/aZ, > 0, i.e., both payoffs increase with respect to the choice of Z,. Since the largest value of Z, from the set of stable strategies corresponds to the linear strategy (see Fig. 5), any lower initial condition (thus leading to a particular nonlinear solution) must lower the payoff of each player. The second claim concerns the differentiability of the value functions. Assume an arbitrary steady state X,
for

X>X,,

= -C(X)/r

Thus the right-hand

(25) for

X>,X,.

limit of the derivatives

(26) of V and W are:

lim V’(X) X-X,+

= 0,

(27)

lim W’(X) X*X,+

= - C’/r = -CX,/r.

(28)

Calculating

the left-hand

derivative

using (16) and (20) yields:

rV’=

-~(~Z-UA)Z’/(~A-~)~=~(CX+~Z)/(~A-

rW’=

-cX-(2A-l)(bZ-aA)Z’/(3A-l)*

= -cX+

(2A - l)(cX+

rZ)/(4A

l),

(29)

- 1).

(30)

Therefore, lim V’(X) x+x,_

=0

and

lim x*x,_

W’(X)

= -cX,/r

cJ

cX+rZ=OoZ’(X,) Differentiability implies that the corresponding which is only satisfied for the linear strategies.

solution

must terminate

=O.

(31)

at Z’ = 0,

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39 (1995) 1215-1236

4.3. Binding resource constraints The properties of nonlinear solutions are important when the resource constraint (6) binds. Linear strategies, which imply asymptotic exhaustion, are infeasible when all resources are used up, because the finite choke price (following from linear demand) implies optimal exhaustion in finite time. As a consequence, a nonlinear solution of the Hamilton-Jacobi-Bellman equations that reaches the steady state at X = R (in finite time) determines the Pareto-efficient outcome when the resource constraint binds, i.e., when R
6 One might argue that the same logic applies when the resource constraint does not bind so that is optimal not to proceed up to the terminal manifold. However, this is not true, because in this case, a solution that satisfies the boundary conditions V(R) = 0 and W(R) = - cR/r cannot support stable strategies when R > - Aar/(bc), see the phase diagram in Fig. 5.

F. Wirl, E. Dockner / European Economic Review 39 (1995) 121.5-1236

Consumer

consumptton(=emialon) timedomain

prices (state space)

“0

X

1231

60

100

time

(1;”

Fig. 7. Consumer prices, consumption ( = emission) and depletion when resource both cases, A = 1 and A = 2; parameters according to Table 1, R = 350.

200

constraints

260

bind in

paths and in particular the date of depletion. At least for this example, the insight of Proposition 3 carries over: the Leviathan is ultimately harmful for the enuironment, i.e., it aduances resource depletion and thus the long-run temperature rise due to global warming.

5. Conclusions Non-competitive and strategic energy producers may retaliate to the threat of carbon taxes by raising the price at front if the government’s purpose is to internalize external costs properly, see Wirl (1994). However, the scope for such preemptions could be considerably weakened if the government were not entirely benevolent but recognized the dual purpose of taxes: to correct for externalities and to raise revenues. Therefore, the objective of this paper is to study the dynamic interactions between energy suppliers (either competitive or monopolistic) and a consumers’ government that is interested in both, the correction of externalities and in the collection of tax revenues. In other words, this government is a blend of the neoclassical benevolent despot and public choice’s Leviathan. The dynamic interactions between the consumers’ government and cartelized suppliers are modelled as an asymmetric differential game; the case of competitive supply is reduced to a standard dynamic optimization problem. For the dynamic

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Economic Review 39 (1995) 1215-1236

game, we derive and compare Nash equilibria in linear and nonlinear Markov strategies, which are subgame perfect. The major finding of this study is that the Leviathan motive lowers (but does not eliminate entirely) the scope for producers’ preemption. Therefore, a partially Leviathan motivated government may improve consumers’ welfare compared with a benevolent government. Hence, having in the real world presumably the choice between no intervention and a Leviathan, the much blamed Leviathan may not be so bad after all. A consequence of the Leviathan is that initial energy taxes are far too high (which is aggravated by cartelized and strategic energy suppliers). Indeed, many ‘Greens’ do not worry much about the government’s underlying motive for taxation as long as external costs are reduced. However, they should be cautious, because calling a Leviathan in order to protect the environment is double-edged: the Leviathan motive lowers the initial emissions through higher taxes (compared with a benevolent government) but increases the long-run environmental harm because such a government is less willing to sacrifice the associated revenues; in short, the above, potential welfare gain entails definitely environmental costs. The reason is that the Leviathan’s taxes fall below their Pigouvian counterpart. This phenomenon - taxes are too low (i.e., not all external costs are internalized) because of revenue maximization - is apparent in other examples as well, presumably in taxing alcohol and cigarettes and in state lotteries where Clotfelter and Cook (1990) document the dominance of the Leviathan motive. Although the Leviathan motive lowers the producers’ price and transfers rents from producers to the consumers, the producers may prefer a government with a stronger preference for tax revenues compared with a moderate Leviathan, because the associated increase in cumulative sales may more than compensate for the lower prices. Finally, market structures (competition versus monopoly) have no impact on cumulative emissions but on the transient behavior: the monopoly lowers initial emissions and the entire transient path of CO, concentrations. This particular finding - the Leviathan motive may lead ultimately to too much cumulative emissions - depends on the linearity of the strategy space. Nonlinear strategies may support any level of cumulative emissions below the linear outcome and thus may lead to excessive conservation over and above the efficient outcome. The reason is that nonlinear strategies follow the linear strategies (prices decline, taxes increase) up to a point where the suppliers refuse to cut prices further and instead increase the price. This triggers a contest over rents where the government retaliates (i.e., it raises taxes even further) so that the demand is effectively choked off at levels (in most cases far) below the steady state that would result from a play in linear strategies. Since nonlinear strategies harm both players, the linear outcome seems more likely. However, nonlinear strategies (with the above qualitative properties) describe the solution when resource constraints limit ultimate carbon emissions. At least for the numerical example in this paper, the conclusion that the Leviathan is harmful for the environment carries over to the case when resource constraints bind.

F. Wirl, E. Dockner/European Economic Review 39 (1995) 1215-1236

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The model considered in this paper captures some of the essential aspects of carbon taxes (welfare and Leviathan motives of governments, fossil energy consumption leads to carbon-dioxide emissions, which increase the CO, concentration in the atmosphere that in turn causes the greenhouse effect) but omits a number of so far known details: global warming is a delayed consequence of greenhouse gas concentration in the atmosphere, non-energy related greenhouse gases, depreciation of greenhouse gas concentrations, abatement, cooperation to manage the global commons of the atmosphere efficiently, etc. The inclusion of these omitted aspects warrants further research, although theoretical results are presumably hard to come by in such an enlarged and thus more realistic framework so that numerical simulations may be necessary.

Acknowledgment We acknowledge helpful comments from two anonymous referees. A first version of the paper was presented at the European Meeting of the Econometric Society at Uppsala, Sweden, August 1993.

Appendix A: Optimal tax strategy for competitive supply We use dynamic programming arguments to derive the optimal tax strategy and define a function W(X,,) that gives the value J’ for X(0) =X0 given. Backward induction yields: IW(X) Solving yields

=my{u(r)

the maximization

+A[T~(T)]

-C(X)

on the right-hand

+ W’(x)f(~)}.

side of (A.l),

(A.1)

using Roy’s identity,

(A~fW’)flf(A-l)f=O=>~~=[a(l-A)-bW’]/[b(2A-l)]. (A.2) Substitution of the optimal tax ( = consumer price rrc ) from (A.21 into (A. 1) leads to a first-order nonlinear differential equation for the value function W(X). We will determine W(X) through guessing a quadratic solution (and not by quadrature), which is possible due to the linear-quadratic specification of the control problem w=

wa + w,x+

+wzx2.

(A.3)

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F. Wirl, E. Dockner/European

Economic Review 39 (1995) 12X5-1236

Equating coefficients yields for the relevant value function does not appear in (A.2)):

coefficients

(the intercept

w,, of the

r-Iv1 = tI(uA -bw,)w,,

(A.4a)

rw2 = -c

(A.4b)

- Obw;,

(A.4c)

&=1/(2A-1).

These equations, using the definition 8, are already arranged in a format that allows for a direct comparison with the monopolistic supply in Section 4. The last and quadratic equation in w2, yields

(A.3 Using that root of (A.5) that induces a stable solution (the first root with the plus before the square root) determines first w1 from (A.4a) and ultimately the optimal tax. The result of these tedious calculations is documented in Proposition 1.

Appendix B: Derivation of a Nash equilibrium in linear Markov strategies The linear strategies can be determined by ‘guessing’ quadratic solutions for the value functions because of the linear-quadratic structure of this dynamic game. We use (A.3) as our ‘guess’ for the value function W(X) of the government and guess a similar quadratic function V(X) for the energy suppliers: V(X)

= ug + u,x+

&x2.

Substitution of the instruments simultaneous equations:

(BJ)

and equating coefficients

rul = 2[ aA - b( Au, + wl)] (Au, + w,)/(3A rv2 = -2b(

Au, + w*)~/(~A

gives the following

- 1)2,

+ (1 - 2A)b(

Au, + w$/(3A

(B.2a) (B.2b)

- l)‘,

(B.2c)

r~~=(1-2A)[b(Av~+w~)-uA](Au,+w~)/(3A-l)~, rw2 = -c

set of

(B.2d)

- 1)‘.

This asymmetrical game, more precisely, the associated system of equations (B.21, can be explicitly solved by using a particular transformation: ’ 21.

.=Av,

22.

.=Au,+w,.

(B.3a)

+ wl,

’ Given the difficulties quite surprising.

(B.3b)

in solving

even such simple asymmetrical

games,

this explicit solution

is

F. Wirl, E. Dockner/European

Economic Reciew 39 (1995) 1215-1236

1235

Substitution of the definitions (B.3) reduces (B.2) to a linear equation system for the unknown coefficients of the value functions, if z, and z2 are indeed known. However, multiplying the first two equations in (B.2) by A and adding the last two equations (B.2c) and (B.2d) yields a system of equations in the new variables zi: T.zl = a(aA rz2 = -c

(B.4a)

- bz,)z,, - abz;,

(B.4b)

CY: = (4A - 1)/(3A

- 1)‘.

(B.4c)

Observe that (B.4) is identical to (A.4) except that the coefficient (Y< 8. In the light of the findings (16) and (B.4), it appears that the solution corresponds to a meta-value function 0: =AV + W. This system (B.4) is easy to solve, (in fact, we just have to replace 8 by (Y in (AS)): z2=

-(rkm)/(2ab),

z,=

-

[

uA(r2-

(B.5a) 2abc)

k d=iiii+(2ab2c).

(BSb)

The knowledge of the coefficients zi is sufficient for the computation of the consumer price 7~ (this follows directly from adding (14) to (15) and substituting the quadratic value functions). The determination of the price and the tax strategy requires to solve (B.2), using (BS), and to substitute the explicit expressions of the coefficients of the quadratic value functions into (14) and (15). These tedious but elementary manipulations allow for a complete analytical characterization of the linear Markov strategies.

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