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Optimal tariffs on exhaustible resources
Journal
of International
Economics
30 (1991) 295-299.
North-Holland
Optimal tariffs on exhaustible
resources
Larry Karp Vniaersity Vniuersiry
of California at Berkeley, Berkeley, C.4 94720, of Southampton, Southampton SO9 .i.vH, VK
VSA.
and
David M. Newbery* Vnicersiry
of Cambridge,
Received October
Cambridge
CB3 9DE.
1989, revised version
L.K
received July 1990
We characterize the Markov perfect equilibria of two games in which oligopsonistic importers of an exhaustible resource confront competitive suppliers who have rational expectations. The games differ only in the timing of moves. or ‘the speed with which participants can adjust their plans. The optimal tariff when sellers move first (are less flexible) differs considerably from that in which buyers move Crst, and sellers retain more control over intertemporal arbitrage opportunities. If the initial stock is small, buyers suffer a disadvantage from being the tirstmover: this is reversed if the stock is large.
1. Introduction
The current price of a competitively supplied exhaustible resource (e.g. oii) in a forward-looking world depends on the level of future demand as well as current demand. An oil-importing country with market power therefore influences the current price by choosing its future levels of imports or import tariffs. This paper characterizes the Markov equilibria of two games in which large importers who behave strategically confront competitive suppliers of a nonrenewable resource. The two games incorporate different assumptions about the ability of exporters to arbitrage supply across time. Newbery (1976) derived the optimal import tariff under the assumptiou that the importer could use an open loop strategy, i.e. was able to announce and precommit to follow a time path of the tariff. If extraction costs are independent of remaining stocks, the optimal unit tariff increases at the rate of interest, so that it has constant present value. This means that the tax does not create a distortion; producers have no incentive to reschedule the time pattern of their supplies, for the tax has the same present cost regardless *This paper
was supported
Information,
and Quantity
Department
of Applied
0022-1996/91/$03.50
0
by the Economic
Signals
Economics
and Social and written on leave from Berkeley.
in Economics,
1991-Elsevier
Science Publishers
Research Council Project on Risk, while Larry Karp was visiting the
B.V. (North-Holland)
286
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Oprimul rar~ji
on exhaustible
resources
of when it is incurred. As the rent on each field must also increase at the rate of interest during the active supply phase, the tariff is levied at a constant rate on the rent; its efficiency can also be seen from its role as a surrogate rent tax. However, as Newbery, and subsequently Kemp and Long (1980) point out, the optimal import tax is dynamically inconsistent. This means that the importer would have an incentive to deviate from the trajectory of taxes announced at the beginning of the program. In the absence of an international agency to enforce that trajectory, it is not a credible announcement as there is no reason for the exporter to believe that it would be carried out. The optimal open loop tax is therefore not a plausible equilibrium outcome. A simple example illustrates the reason for the dynamic inconsistency of the optimal program. Consider the case in which demand for oil falls to zero at some choke price, @,and a dominant importer chooses his tariff given that the remaining oil-importing countries behave competitively. At some date the domestic tariff-inclusive price of imported oil in the dominant importing country will have risen to fi. but the world price of oil will still be below that price, so the rest of the world will still be producing and buying oil. At that point it would clearly be desirable for the dominant importer to depart from its optimal import plan, and resume oil imports, since their cost will be less than their value. The reason for announcing the earlier tariff plan was to drive the world price of oil down in the early stages, and thereby derive the benefits of market power. Now that those benefits have been enjoyed, it would be convenient to depart from the earlier announced plan. The noncooperative Nash equilibria in the two games studied in this paper are subgame perfect, unlike the optimal program described above. The state variable in each game is the remaining stock of the resource. Perfection means that the proposed equilibrium is an equilibrium for any possible subgame, i.e. for any possible level of the stock. Whether or not agents ‘make a mistake’, or for some other reason deviate from their equilibrium strategies, the continuation of those strategies represents equilibrium behaviour. Perfection is stronger than time consistency, since the latter requires only that the continuation of the proposed equilibrium represents an equilibrium when no agent has deviated in the past. The distinction between time consistency and perfection in the context of resource models was noted by Reinganum and Stokey (1985) and is discussed in Karp and Newbery (1989a). The model in the next section assumes that competitive exporters with rational expectations choose their supply in each period before the importers choose their tariffs. Section 2 considers the case where importers choose tariffs before exporters make their supply decisions. When we wish to distinguish the equilibrium tariffs in the two models, we will refer to the former as ‘PEMF’ (perfect, exporters move first) and the latter as ‘PIMF (perfect, importers move tirst). Both models use a continuous time formula-
L.. Karp
and D..M. Newbery,
Optimal
rurijfs
on exhaustible
resources
187
tion, so they should be interpreted as the limiting case where the length of each period tends to zero. Although it is natural to identify the ‘first-mover’ as the more sophisticated agent(s), this may not be appropriate. If any agent chooses a nonequilibrium move in a given period, the other agents would like to change their decisions. However, there is typically some cost in making unplanned changes; these costs may, in the short run, inhibit an agent from responding to nonequilibrium behaviour. The two games described above embody different assumptions regarding the relative costs of making unexpected changes in either the tariff or the level of supply. The first game, in which exporters move first, corresponds to the situation where short-run supply decisions are not flexible, so that arbitrage opportunities are limited. In this case, a change in the current tariff(s) changes the pattern of current demand and the current market-clearing price, but not the level of aggregate supply. Of course, if importers were to deviate from their equilibrium tariff, the competitive exporters would want to arbitrage supply across time in order to maximize profits. If producers can instantaneously revise their production plans and vary the amount of oil they sell and the amount they retain in the ground for future extraction and sale, then the second model, in which importers move first, is appropriate. The two models represent limiting cases regarding the supply flexibility of producers (and thus their ability to arbitrage), relative to the ability of importers to make unanticipated policy changes. Numerical methods are used to compare the price trajectories and welfare under the two equilibria. We show that an apparently slight difference in the timing of moves makes a profound difference to the optimal tariff and the intertemporal equilibrium. 2. Exporters move first The supply and demand of oil is modelled as a dynamic game. The nature of the game depends on the information upon which the large importing countries and competitive suppliers base their decisions. This section studies the equilibrium when exporters choose supply in each period before importers announce their tariffs. We define S, as the remaining stock at time t, Q, as the exporters’ total rate of extraction (=sales) and r, as the n-dimensional vector of unit tariffs for the n importing countries. We assume that demand is stationary and restrict attention to Markov perfect equilibria. The Markov assumption means that agents base their decisions only on information that has direct economic relevance and is currently observable. In the present context, the state of the game is described by the remaining stock of oil, S, so the Markov assumption means that the equilibrium supply is given by some function Q(S), and the equilibrium tariff is given by a function r(S,Q). The equilibrium tariff can depend on Q, but not vice versa,
258
L. Karp
and D..Cf. Newlbery,
Optimal
tarifrs
on exhaustible
resources
because exporters move first in each period (in the limit at every instant). The Markov assumption is useful because it often leads to a unique equilibrium. Without that, or a similarly strong assumption, there typically exists a continuum of equilibria based on trigger, or ‘punishment’ strategies [see, for example Olsen (1989)]. The Markov assumption also seems plausible, since it means that the outcome of the game is insensitive to agents’ mistakes, or deviations which arise for other reasons. The equilibrium price received by the exporters is p=p[Q(S), r(S, Q)] =p(S). Since exporters are competitive, they take as given the current price and the trajectory of price, but they understand the dependence of these on the current stock, S,. The sellers are individually small, so they do not try to manipulate the rate of supply or the future value of the stock as a means of affecting the tariffs. If extraction costs are constant, c, the equilibrium must satisfy the Hotelling arbitrage relationship:
dp
-$=r(P-c),
(1)
where r is the rate of interest. The Markov assumption means that the importers have no incentive to use their current tariffs to affect future supply: given the current values of S and Q, the future stock, and thus the future rate of supply is beyond their control. Therefore, at each point in time importer i chooses a tariff to maximize the instantaneous flow of domestic welfare (=consumer surplus+ tariff revenues), taking as given Q, and the rivals’ tariffs, rei. Karp and Newbery (1989b) derive the Nash equilibrium unit tariff, $, for this static problem as
(2)
where qi is demand by country i. We assume that the Nash tariff of the static one-shot game, obtained when importers choose tariffs and face fixed aggregate supply, is unique; this is true for linear demand, and is also true more generally. Under this assumption, a straightforward proof by contradiction shows that any Markov tariff in the dynamic game (where exporters move first) depends on the current flow of aggregate supply (since markets clear), but not on the stock. The basis of the argument is the fact that no importer is able to influence either the current or future flow of supply. Any tariff rule that was a nontrivial function of stock would violate either the assumption of Markov perfection, or the assumption of uniqueness of the
L. Karp and D.M. Newbery. Optimal tariffs on exhaustible resources
‘89
Nash equilibrium to the static game. Consequently, the tariff given by eq. (2) is the unique Markov equilibrium of this game. When the demand schedule is linear, demand by i is qi = r’/?(p- p - ?), where ri is the market share of country i when no tariffs are used, b is a scaling parameter, and the aggregate (untaxed) demand schedule is Q=P(p--p). The formula for the Nash equilibrium unit tariff is then T’=a’(p-p),
(3)
when p is the current price. For the case of the linear demand aggregate demand are qi=a’(l
schedule, importer
i’s demand
and
-@(p-p),
(4)
The second line of (4) uses (1) to write p,=(j-~)e”‘-~)+c, where T is the amount of time during which extraction is positive, so T-t is the time remaining (at t) before the oil is exhausted; T can be found from the exhaustion condition. If So is the initial stock of oil, then the time integral of aggregate consumption from 0 to T is equal to this stock, whence the value of T and hence p. can be derived:
The term [l -Z(ai)2] measures the effect of the exercise of market power by the importers. This can be written as 1 -H, where H is the Herfindahl index. If all n importing countries are identical, so that ai= l/n, and if the stock is such that (in equilibrium) there are T years to go before exhaustion, then the equilibrium value of the tariff is
T=t(l-e-“),
AE~-c,
(64
and the equilibrium ratio of the tariff to the current rent is
---=~(err-l). T
p-c
n
(6W
Both of these expressions can be expressed as a function of current stock, S,
290
L. Karp
and D.M.
.Vrwbery,
Optimal
tariffs
on exhaustible
resouwes
by noting that the equilibrium value of T is a monotonic function of S, (i.e. current stock). When importers are identical and p= 1, eq. (5) for S becomes
s= +[T-$(T)I.
$(r)=je-rUdil.
For purposes of comparison with the tariff studied in the next section, it is useful to express the equilibrium value of this PEMF tariff as a function of the remaining stock. However, note that this representation gives the equilibrium value of that tariff. The PEMF equilibrium strategies are functions of the rate of extraction, Q; in equilibrium Q is a function of S. If the exporters choose a nonequilibrium extraction rate, the tariff given by (6a) and (5a) would not be equilibrium responses.
3. Importers move first This section derives the (noncooperative Nash) Markov perfect equilibrium for the case where importers simultaneously choose their policies in the current period before exporters choose aggregate supply. This situation is considerably more complicated than the previous model, where the timing of moves implies that the only strategic play is amongst importers, who play a sequence of static games. Those static games are identical in every period (instant), except for the changing aggregate supply. In the model studied here each importer is able to affect the current rate of extraction by means of its tariff. Therefore the game amongst importers is truly dynamic, since they need to take into account how their effect on current sales will alter the future stock, and thus alter their rivals’ future tariffs. In the model studied here, the equilibrium vector of tariffs is given by a function r(S) and the equilibrium rate of extraction is given by a function Q(&T). The derivation of the equilibrium conditions for the continuous time PIMF equilibrium is given in a working paper [Karp and Newbery (1989c)]. We sketch the derivation here. We also comment on the fact that the equilibrium does not depend on whether agents choose unit or ad valorem tariffs, or import quantities. This result holds in the continuous time model, but does not hold in static models, or in the discrete time dynamic model. This limiting result implies that the form that policy intervention takes in strategic settings may be less significant than is commonly thought. We begin with a discrete-time game in which agents make decisions at intervals of length E. We assume that the value functions and the price function determined by equilibrium behaviour are analytic in E (and in the
stock of the resource), so that the limiting equilibrium as E+O lies near the equilibria for small positive values of E, and so that it is legitimate to take first-order Taylor expansions of those functions. Given the restriction to (smooth) Markov equilibria, the discrete stage can be characterized using dynamic programming. The solution to the game involves endogenous value functions, which give the pay-off to an importer as a function of the current stock, and an endogenous price function. The latter gives the price at each stage as a function of the stock size and the levels of policy variables that players use at that stage. In equilibrium, importers’ current policies are functions of the current stock, so in equilibrium the current price is a function of only the current stock. Importers are free to use nonequilibrium policies in the present, but they cannot commit themselves to the use of such policies in the future, in light of the requirement of perfection. The dependence of the current price on the current policies is of the same order of magnitude as E. This fact stems from the exporter’s ability to arbitrage, and from the Markov assumption, which means that the exporter expects future import policies to depend on future values of the state variable. If E, the length of the current period, is small, then anything an importer chooses to do in the current period (i.e. any type of nonequilibrium behaviour) has only a small effect on the trajectory of future behaviour, and can therefore have only a small effect on the current price (which depends on future behaviour). As E+O, the importers become unable to alter the current price, but this does not imply the demise of their market power. The reason is that their current actions have a first-order effect on the current rate of sales, and thus on the evolution of the state, and thus on the trajectory of future prices. The fact that, in the limit, importers cannot alter the current price does, however, have an important implication. It means that the equilibrium does not depend on whether each importer chooses a state-dependent tariff rule, or a state-dependent import quantity. In static models, or in the discrete stage dynamic model, the equilibrium depends on the choice of the policy instrument. In such models, this dependence is owing to the fact that agents face different optimization problems, and thus make different decisions, depending on what aspect of their rivals’ behaviour they take as given. For example, if importers choose import quantities, then other importers act (under the assumption of Nash behaviour) as if they are unable to affect those quantities in the current stage. If, on the other hand, agents choose tariffs, then the fact that the current price depends on the current level of policy variables (for E>O) is important. Agent i knows that his current policy will affect the current world price, and thus affect his rivals’ import levels at the current stage. Agent i’s optimization problem is consequently different when he takes as given his rivals’ tariffs rather than import quantities. The same type of argument applies if the comparison is between unit and ad
292
L. Karp
and D..tl.
Newbery,
Optimal
rarqfj
on exhaustible
resources
valorem tariffs. As we noted above, in the limit as s--r0 the dependence of current price on current policy levels disappears; therefore in the limit there can be no dependence of the equilibrium on the choice of policy variables, at least for the reason described above. One might ask whether there is some subtler form of dependence, owing perhaps to dynamic interactions. However, future behaviour depends only on future values of the state (the Markov assumption again), so the choice of policy instruments could only affect future outcomes if it were able to affect the evolution of the state. This requires that the choice of policy instruments affect the equilibrium rate of extraction at a point in time; but, as we have just argued, for an arbitrary level of the stock, the equilibrium rate of extraction at any point in time is independent of the choice of policy instruments (in the limit as E-PO). The simplest case to analyze is one in which all n importers are identical. The price at any date is a function of the stock, P(S); aggregate equilibrium demand of importers other than i is some function y(S). For the case of linear demand [i.e. Q=fi(P--p)] and constant costs, the instantaneous level of welfare (in cash terms) of a country is [j7-P(S)-nq/2]q, where I/n is the (competitive) share of the country, and /? in the demand equation has been set to unity by choice of units. The dynamic programming equation for a representative importer is
rJ(S)=m;x{[p-P(S)-
y]q-J&(S)+q]]
where q is the importer’s rate of consumption value of S. The first-order condition is D--nq-P(S)-J,=O,
‘7
and J,=dJ/dS
(7)
is his shadow
(8)
which says that the difference between domestic and world price (the unit tariff) equals the importer’s shadow value of the resource.’ Eq. (8) has two important implications, neither of which depends on the linear functional form. When consumption is positive the importer’s consumer surplus is positive: the importer’s present discounted utility would be increased by a small increase in the stock, so the shadow value of the
‘The same first-order condition holds for the (dynamically inconsistent) open loop tariff [Karp and Newbery (1989b)], but the shadow values in the two cases obey different equations of motion; thus, the open loop tariB and the perfect tariff considered here are different.
L. Karp
and D.JI.
5rwbery.
Optimal
tar@
on exhaustible
resources
293
resource is positive. The first implication of (8), then, is that the tariff is positive on the equilibrium path, provided that consumption is positive and the stock is finite. The second implication is that the equilibrium tariff approaches zero as the stock of the resource approaches zero or becomes infinite. Therefore, for sufficiently large resource stocks the equilibrium tariff first increases and then decreases over time. To see that the tariff approaches zero as the stock approaches zero, note that the price received by the importer must approach the choke price, j, as exhaustion approaches; if this were not the case the price function would be discontinuous at S=O.* The economic interpretation is that as the stock decreases, aggregate supply (a flow) must approach zero. As the aggregate supply approaches zero, so must the excess supply facing any importer. Therefore the elasticity of excess supply becomes infinite, and it becomes optimal to impose a zero tariff. As the stock becomes infinite the scarcity value of the resource approaches zero; that is, the resource becomes identical to a standard product that can be produced at constant cost. Under these conditions the excess supply curve facing any importer is perfectly elastic, and the optimal tariff is again zero. To complete the derivation, substitute nq=Q into (8) to obtain: j5-P(S)-J,=nq=Q=
-dS/dt.
(9)
Then use nq= y+q and (9) in (7) and rearrange to obtain:
rJ=
P-p-O2n
l)Js -
(p-p-
Jd
(10)
Eq. (1) describes the response of suppliers to any deviations from the planned profile of imports, and as p= P(S), dpfdt = P,dSfdt. Replacing dS/dt by -Q, given by (9), enables us to rewrite (1) (after some rearrangement) as
J
_dP-4 S----+j-p. Ps
(11)
*There cannot be an upward jump in price at the exhaustion time, since this would violate the sellers’ arbitrage condition; if there was a downward jump in price at the exhaustion time, there would be a linal interval during which time the importers were paying more than their choke price, which would not be optimal.
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Newbery,
Optimal
tariffs on exhaustible resources
Prices 10
Tariff/rem
(log)
r
6-
Tariff/rent
(RHS)
2-
0, 0.125
,
II 0.5
1
II
I
2
I
I
8
I
I
/I
0.001
128
32
Stock (log scale) Fig. 1. Tariffs as a function
of stock (n=3,r=59,). Key: PEMF price; - - - PEMF
PlMF t/rent.
price; ..........
PIMF
t/rent;
Eqs. (10) and (11) comprise a system of ordinary differential equations in S, with boundary conditions P(0) =p and J(0) =O.j 4. Comparing the equilibrium tariffs It is relatively easy to study the comparative statics of the PEMF equilibrium tariff. The tariff is monotonic in the remaining stock, S; as S increases, the level of the tariff tends to some finite amount, A/n; as the stock approaches zero, the tariff falls to zero. The ratio of the tariff to the rent tends to infinity as the stock increases. As the rate of discount, r, increases, the ratio of the tariff to the rent increases. The comparative statics of the PIMF equilibrium tariff are more complicated, and hence we resort to numerical calculations, whose results are presented in the following graphs. Fig. 1 graphs the prices and the ratios of tariff to rent for the two -‘Eq. (10) implies that J,(O)=O, and (I I) implies that P,(O)= 30. The in&trite derivative of ps makes the system of equations ‘stilf’ and requires rather delicate techniques. The pair of ordinary differential equations were integrated numerically using the NAG Fortran Library Routine DOZEBF, using a variable-order, variable-step method of implementing the Backward Differentiation Formulas described in Hall and Watt (1976). The output was read into a PC spreadsheet program to produce the graphics, which are accurate plots.
L. Karp
and D.M.
Nrwbery,
Optimal
tar$fs
on exhaustible
resources
295
Tariff levels 3.5
2
8
32
128
Stock (log scale) Fig. 2. Tariff levels as discount rate varies (n=3). Key: PIMF 5%; PEMF 5%; - - - PEMF 15%.
PIMF 15%; .........
equilibria as functions of remaining stock (on a log scale), using base parameters jj= 10, c= 1, I= 5 percent, and n=3 countries.4 The ratio of the tariff to rent is read off the right-hand scale, which is logarithmic. The ratio of tariff to rent increases monotonically with stock; we showed analytically that this holds under PEMF. Fig. 2 shows the sensitivity of the results to changes in the discount rate, r. For high stock levels, an increase in the discount rate decreases the equilibrium PIMF tariff; as the stock is run down the relationship is reversed. This result illustrates that the comparative statics of the PIMF tariff are ambiguous, since they depend on the stock level. The PEMF tariff increases with the discount rate, as predicted by the analytic results reported above. The welfare properties of the two tariffs are quite different. For low stock ‘It is natural to compare the tariff with the rent, as the optimal precommitted open loop tariff has a constant ratio to the rent, and the rent represents the ‘taxable surplus’ that might reasonably be appropriated by the importers, especially if acting in concert. ‘See Bergstrom (1981) and Bergstrom, Cross and Porter (1981).
296
L. Karp and D.M. Newbery, Optimal tariffs on exhaustible resources PEMF/PIlMF 1.35
welfare
I
------------------__
l---------
I
0.95 0.125
1 0.5
I
I 2
I
I,
I 1
8
32
I
1 128
Stock (log scale) Fig. 3. Importer
welfare
as n varies (ratio of PIMF to PEMF importers; eight importers.
welfare,
r=57;).
Key: -
two
levels the importers’ welfare is higher under PEMF. ,Fig. 3 shows that importers gain approximately 9 percent under PEMF (relative to PIMF) when there are eight buyers; the gain is approximately 30 percent when there are two buyers. This comparison is quite insensitive to changes in stock levels, provided that the stock is not large. Being the ‘first-mover is a disadvantage to the importers in this situation. However, for very large stock levels, buyers do better under PIMF than under PEMF. This result can be explained by the following argument. We noted above that as the stock level approaches infinity, the buyers’ shadow value of the resource, and hence the PIMF tariff, approaches zero; the PEMF tariff, on the other hand, converges to A/n>O. In the limit as S+cc the model collapses to that of a reproducible good, produced at constant cost c. In that case the aggregate supply function is perfectly elastic and the optimal tariff equals zero, which is the equilibrium (limiting) PIMF tariff. That is, in the limiting case the PIMF tariff maximizes joint buyer welfare (as well as social surplus), and hence maximizes the welfare of importer i. The limiting PEMF tariff is positive, however. The
L. Karp
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rari/fi
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remmes
297
assumption that exporters move first means that importers take the aggregate supply function as perfectly inelastic. and use positive tariffs in equilibrium.’ The above argument relies on a comparison of the importers’ flow of welfare in the two games, in the limit as S-+x). Numerical experiments confirm that the result also holds for large but finite stock levels. For large stock levels, a high discount rate increases the likelihood that buyers do better under PIMF, relative to PEMF. The reason is that for large stock levels the games ‘are close to’ the limiting games described above. and in those games the flow of welfare is higher under PIMF than PEMF: and for high discount rates the present value of the stream of welfare is dominated by the flow near the beginning of the program (while the stock is large). The previous comments imply that if the stock levels and the discount rate are sufhciently large, the PEMF tariff is disadvantageous.6 This means that importers would have a higher present value of welfare if they could commit to using a zero tariff throughout the program. The possibility of disadvantageous market power was noted by Maskin and Newbery (1978, revised 1990) in the context of a two-period nonrenewable resource model. That model assumed a single dominant importer facing competitive producers and competing buyers; all buyers behave strategically in our model. Maskin and Newbery study the situation where the dominant importer moves first in each period, so the natural comparison is with the PIMF tariff. Owing to the strategic symmetry of buyers in our model, however, we do not find any situation where the PIMF tariff is disadvantageous.
5. Conclusions
We have derived and characterized the Markov perfect equilibria for two games in which dominant buyers use tariffs to exert market power when confronted by competitive producers. The two games embody different assumptions regarding the ability of producers to arbitrage supply across time. In neither game do producers have any incentive to change their behaviour in equilibrium. If importers move first .- the PIMF assumption exporters would be able to change the level of aggregate exports in response ‘This behaviour does not reflect irrationality or myopia on the part of any agent; it is simply a consequence of the Markov assumption and of the order of play. The Markov assumption means that current buyers’ behaviour cannot affect the sellers’ future decisions. Therefore, the buyers cannot do better than to maximize their welfare in the current period; Karp (1987) obtains a similar result for the case of a single buyer. 6For large stock levels, when the supply is nearly perfectly elastic, a zero tariff results in a higher flow of welfare to importers than does a tariff that is close to A/n. For high discount rates, any comparison of payoffs under dilferent equilibria depends on the comparison of the flow of welfare near the beginning of the trajectory.
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to a deviation from equilibrium on the part of importers; they lack this ability if they must move first - the PEMF assumption. The equilibrium PEMF tariff is a monotonic function of time, but for large initial stocks the PIMF tariff first increases then decreases. The PEMF tariff is greater than the PIMF tariff, for given stock levels. Provided that the initial stock is not too large, importers do better under PEMF; that is, there is a first-mover disadvantage. This is reversed for very large stock levels. If the stock level is sufficiently large, market power is disadvantageous under the PEMF assumption. In addition to the specific results described below, this paper illustrates techniques that are useful in studying imperfectly competitive resource markets. The most important advantage of these methods is that they permit the study of perfect equilibria; previous papers have concentrated on open loop equilibria or used two-period models. There are a number of promising directions for future research. It would be useful to generalize the model to the case of asymmetric buyers, in order to study the situation where there are both competitive and oligopsonistic buyers in the market. This would be straightforward. It would also be interesting to’ incorporate noncompetitive sellers. This is a more challenging problem, since it would require a multi-dimensional state vector, consisting of the stock levels of all sellers.
References Bergstrom, Theodore, 1981, On capturing oil rents with national excise tax, American Economic Review 71, 194201. Bergstrom, Theodore, John Cross and Richard Porter, 1981, Efiiciency inducing taxation for a monopolistically supplied depletable resource, Journal of Public Economics 15, no. 1, 23-32. Hall, G. and J.M. Watt, eds., 1976, Modern numerical methods for ordinary differential equations (Clarendon Press, Oxford). Karp, Larry, 1987, Consistent tariffs with dynamic supply response, Journal of International Economics 23, 369-376. Karp, Larry and David Newbery, 1989a, Intertemporal consistency issues in depletable resources, Discussion Paper no. 346 (Centre for Economic Policy Research, London), in: James Sweeney and Allen Kneese, eds., Handbook of natural resource and energy economics, Vol III (North-Holland, Amsterdam), forthcoming. Karp, Larry and David Newbery, 1989b, Time consistent oil import tariffs, Discussion Paper no. 344 (Centre for Economic Policy Research, London). Karp, Larry and David Newbery, 1989~. Optimal tariffs on exhaustible resources, Economic Theory Discussion Paper no. 141 (Department of Applied Economics, Cambridge, England). Kemp, Murray and Ngo Long, 1980, Optimal tariffs and exhaustible resources, in: Murray Kemp and Ngo Long, eds., Exhaustible resources, optimality and trade, Essay 18 (NorthHolland, Amsterdam). Maskin, Eric and David Newbery, 1978, Rational expectations with market power: The paradox of the disadvantageous monopolist, Warwick Economic Discussion Paper and Massachusetts Institute of Economics Discussion Paper (Cambridge, MA). Maskin, Eric and David Newbery, 1990, Disadvantageous oil tariffs and dynamic consistency, American Economic Review 80, no. 1, 143-156.
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and D.M.
Newbery,
Optimul
tariffs on exhaustible
resources
299
Newbery. David, 1976, A paradox in tax theory: Optimal tarilTs on exhaustible resources, SEER Technical Paper (Stanford Economics Department). Olsen, Trond, 1989, A folk theorem for rent extracting tariffs on exhaustible resources. Unpublished working paper (Norwegian Center for Research in Organization and Management, Bergen. Norway). Reinganum. Jennifer and Nancy Stokey, 1985, Oligopoly extraction of a common property natural resource: The importance of the period of commitment in dynamic games, International Economic Review 26 (I), 161-173.
of International
Economics
30 (1991) 295-299.
North-Holland
Optimal tariffs on exhaustible
resources
Larry Karp Vniaersity Vniuersiry
of California at Berkeley, Berkeley, C.4 94720, of Southampton, Southampton SO9 .i.vH, VK
VSA.
and
David M. Newbery* Vnicersiry
of Cambridge,
Received October
Cambridge
CB3 9DE.
1989, revised version
L.K
received July 1990
We characterize the Markov perfect equilibria of two games in which oligopsonistic importers of an exhaustible resource confront competitive suppliers who have rational expectations. The games differ only in the timing of moves. or ‘the speed with which participants can adjust their plans. The optimal tariff when sellers move first (are less flexible) differs considerably from that in which buyers move Crst, and sellers retain more control over intertemporal arbitrage opportunities. If the initial stock is small, buyers suffer a disadvantage from being the tirstmover: this is reversed if the stock is large.
1. Introduction
The current price of a competitively supplied exhaustible resource (e.g. oii) in a forward-looking world depends on the level of future demand as well as current demand. An oil-importing country with market power therefore influences the current price by choosing its future levels of imports or import tariffs. This paper characterizes the Markov equilibria of two games in which large importers who behave strategically confront competitive suppliers of a nonrenewable resource. The two games incorporate different assumptions about the ability of exporters to arbitrage supply across time. Newbery (1976) derived the optimal import tariff under the assumptiou that the importer could use an open loop strategy, i.e. was able to announce and precommit to follow a time path of the tariff. If extraction costs are independent of remaining stocks, the optimal unit tariff increases at the rate of interest, so that it has constant present value. This means that the tax does not create a distortion; producers have no incentive to reschedule the time pattern of their supplies, for the tax has the same present cost regardless *This paper
was supported
Information,
and Quantity
Department
of Applied
0022-1996/91/$03.50
0
by the Economic
Signals
Economics
and Social and written on leave from Berkeley.
in Economics,
1991-Elsevier
Science Publishers
Research Council Project on Risk, while Larry Karp was visiting the
B.V. (North-Holland)
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on exhaustible
resources
of when it is incurred. As the rent on each field must also increase at the rate of interest during the active supply phase, the tariff is levied at a constant rate on the rent; its efficiency can also be seen from its role as a surrogate rent tax. However, as Newbery, and subsequently Kemp and Long (1980) point out, the optimal import tax is dynamically inconsistent. This means that the importer would have an incentive to deviate from the trajectory of taxes announced at the beginning of the program. In the absence of an international agency to enforce that trajectory, it is not a credible announcement as there is no reason for the exporter to believe that it would be carried out. The optimal open loop tax is therefore not a plausible equilibrium outcome. A simple example illustrates the reason for the dynamic inconsistency of the optimal program. Consider the case in which demand for oil falls to zero at some choke price, @,and a dominant importer chooses his tariff given that the remaining oil-importing countries behave competitively. At some date the domestic tariff-inclusive price of imported oil in the dominant importing country will have risen to fi. but the world price of oil will still be below that price, so the rest of the world will still be producing and buying oil. At that point it would clearly be desirable for the dominant importer to depart from its optimal import plan, and resume oil imports, since their cost will be less than their value. The reason for announcing the earlier tariff plan was to drive the world price of oil down in the early stages, and thereby derive the benefits of market power. Now that those benefits have been enjoyed, it would be convenient to depart from the earlier announced plan. The noncooperative Nash equilibria in the two games studied in this paper are subgame perfect, unlike the optimal program described above. The state variable in each game is the remaining stock of the resource. Perfection means that the proposed equilibrium is an equilibrium for any possible subgame, i.e. for any possible level of the stock. Whether or not agents ‘make a mistake’, or for some other reason deviate from their equilibrium strategies, the continuation of those strategies represents equilibrium behaviour. Perfection is stronger than time consistency, since the latter requires only that the continuation of the proposed equilibrium represents an equilibrium when no agent has deviated in the past. The distinction between time consistency and perfection in the context of resource models was noted by Reinganum and Stokey (1985) and is discussed in Karp and Newbery (1989a). The model in the next section assumes that competitive exporters with rational expectations choose their supply in each period before the importers choose their tariffs. Section 2 considers the case where importers choose tariffs before exporters make their supply decisions. When we wish to distinguish the equilibrium tariffs in the two models, we will refer to the former as ‘PEMF’ (perfect, exporters move first) and the latter as ‘PIMF (perfect, importers move tirst). Both models use a continuous time formula-
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tion, so they should be interpreted as the limiting case where the length of each period tends to zero. Although it is natural to identify the ‘first-mover’ as the more sophisticated agent(s), this may not be appropriate. If any agent chooses a nonequilibrium move in a given period, the other agents would like to change their decisions. However, there is typically some cost in making unplanned changes; these costs may, in the short run, inhibit an agent from responding to nonequilibrium behaviour. The two games described above embody different assumptions regarding the relative costs of making unexpected changes in either the tariff or the level of supply. The first game, in which exporters move first, corresponds to the situation where short-run supply decisions are not flexible, so that arbitrage opportunities are limited. In this case, a change in the current tariff(s) changes the pattern of current demand and the current market-clearing price, but not the level of aggregate supply. Of course, if importers were to deviate from their equilibrium tariff, the competitive exporters would want to arbitrage supply across time in order to maximize profits. If producers can instantaneously revise their production plans and vary the amount of oil they sell and the amount they retain in the ground for future extraction and sale, then the second model, in which importers move first, is appropriate. The two models represent limiting cases regarding the supply flexibility of producers (and thus their ability to arbitrage), relative to the ability of importers to make unanticipated policy changes. Numerical methods are used to compare the price trajectories and welfare under the two equilibria. We show that an apparently slight difference in the timing of moves makes a profound difference to the optimal tariff and the intertemporal equilibrium. 2. Exporters move first The supply and demand of oil is modelled as a dynamic game. The nature of the game depends on the information upon which the large importing countries and competitive suppliers base their decisions. This section studies the equilibrium when exporters choose supply in each period before importers announce their tariffs. We define S, as the remaining stock at time t, Q, as the exporters’ total rate of extraction (=sales) and r, as the n-dimensional vector of unit tariffs for the n importing countries. We assume that demand is stationary and restrict attention to Markov perfect equilibria. The Markov assumption means that agents base their decisions only on information that has direct economic relevance and is currently observable. In the present context, the state of the game is described by the remaining stock of oil, S, so the Markov assumption means that the equilibrium supply is given by some function Q(S), and the equilibrium tariff is given by a function r(S,Q). The equilibrium tariff can depend on Q, but not vice versa,
258
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on exhaustible
resources
because exporters move first in each period (in the limit at every instant). The Markov assumption is useful because it often leads to a unique equilibrium. Without that, or a similarly strong assumption, there typically exists a continuum of equilibria based on trigger, or ‘punishment’ strategies [see, for example Olsen (1989)]. The Markov assumption also seems plausible, since it means that the outcome of the game is insensitive to agents’ mistakes, or deviations which arise for other reasons. The equilibrium price received by the exporters is p=p[Q(S), r(S, Q)] =p(S). Since exporters are competitive, they take as given the current price and the trajectory of price, but they understand the dependence of these on the current stock, S,. The sellers are individually small, so they do not try to manipulate the rate of supply or the future value of the stock as a means of affecting the tariffs. If extraction costs are constant, c, the equilibrium must satisfy the Hotelling arbitrage relationship:
dp
-$=r(P-c),
(1)
where r is the rate of interest. The Markov assumption means that the importers have no incentive to use their current tariffs to affect future supply: given the current values of S and Q, the future stock, and thus the future rate of supply is beyond their control. Therefore, at each point in time importer i chooses a tariff to maximize the instantaneous flow of domestic welfare (=consumer surplus+ tariff revenues), taking as given Q, and the rivals’ tariffs, rei. Karp and Newbery (1989b) derive the Nash equilibrium unit tariff, $, for this static problem as
(2)
where qi is demand by country i. We assume that the Nash tariff of the static one-shot game, obtained when importers choose tariffs and face fixed aggregate supply, is unique; this is true for linear demand, and is also true more generally. Under this assumption, a straightforward proof by contradiction shows that any Markov tariff in the dynamic game (where exporters move first) depends on the current flow of aggregate supply (since markets clear), but not on the stock. The basis of the argument is the fact that no importer is able to influence either the current or future flow of supply. Any tariff rule that was a nontrivial function of stock would violate either the assumption of Markov perfection, or the assumption of uniqueness of the
L. Karp and D.M. Newbery. Optimal tariffs on exhaustible resources
‘89
Nash equilibrium to the static game. Consequently, the tariff given by eq. (2) is the unique Markov equilibrium of this game. When the demand schedule is linear, demand by i is qi = r’/?(p- p - ?), where ri is the market share of country i when no tariffs are used, b is a scaling parameter, and the aggregate (untaxed) demand schedule is Q=P(p--p). The formula for the Nash equilibrium unit tariff is then T’=a’(p-p),
(3)
when p is the current price. For the case of the linear demand aggregate demand are qi=a’(l
schedule, importer
i’s demand
and
-@(p-p),
(4)
The second line of (4) uses (1) to write p,=(j-~)e”‘-~)+c, where T is the amount of time during which extraction is positive, so T-t is the time remaining (at t) before the oil is exhausted; T can be found from the exhaustion condition. If So is the initial stock of oil, then the time integral of aggregate consumption from 0 to T is equal to this stock, whence the value of T and hence p. can be derived:
The term [l -Z(ai)2] measures the effect of the exercise of market power by the importers. This can be written as 1 -H, where H is the Herfindahl index. If all n importing countries are identical, so that ai= l/n, and if the stock is such that (in equilibrium) there are T years to go before exhaustion, then the equilibrium value of the tariff is
T=t(l-e-“),
AE~-c,
(64
and the equilibrium ratio of the tariff to the current rent is
---=~(err-l). T
p-c
n
(6W
Both of these expressions can be expressed as a function of current stock, S,
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L. Karp
and D.M.
.Vrwbery,
Optimal
tariffs
on exhaustible
resouwes
by noting that the equilibrium value of T is a monotonic function of S, (i.e. current stock). When importers are identical and p= 1, eq. (5) for S becomes
s= +[T-$(T)I.
$(r)=je-rUdil.
For purposes of comparison with the tariff studied in the next section, it is useful to express the equilibrium value of this PEMF tariff as a function of the remaining stock. However, note that this representation gives the equilibrium value of that tariff. The PEMF equilibrium strategies are functions of the rate of extraction, Q; in equilibrium Q is a function of S. If the exporters choose a nonequilibrium extraction rate, the tariff given by (6a) and (5a) would not be equilibrium responses.
3. Importers move first This section derives the (noncooperative Nash) Markov perfect equilibrium for the case where importers simultaneously choose their policies in the current period before exporters choose aggregate supply. This situation is considerably more complicated than the previous model, where the timing of moves implies that the only strategic play is amongst importers, who play a sequence of static games. Those static games are identical in every period (instant), except for the changing aggregate supply. In the model studied here each importer is able to affect the current rate of extraction by means of its tariff. Therefore the game amongst importers is truly dynamic, since they need to take into account how their effect on current sales will alter the future stock, and thus alter their rivals’ future tariffs. In the model studied here, the equilibrium vector of tariffs is given by a function r(S) and the equilibrium rate of extraction is given by a function Q(&T). The derivation of the equilibrium conditions for the continuous time PIMF equilibrium is given in a working paper [Karp and Newbery (1989c)]. We sketch the derivation here. We also comment on the fact that the equilibrium does not depend on whether agents choose unit or ad valorem tariffs, or import quantities. This result holds in the continuous time model, but does not hold in static models, or in the discrete time dynamic model. This limiting result implies that the form that policy intervention takes in strategic settings may be less significant than is commonly thought. We begin with a discrete-time game in which agents make decisions at intervals of length E. We assume that the value functions and the price function determined by equilibrium behaviour are analytic in E (and in the
stock of the resource), so that the limiting equilibrium as E+O lies near the equilibria for small positive values of E, and so that it is legitimate to take first-order Taylor expansions of those functions. Given the restriction to (smooth) Markov equilibria, the discrete stage can be characterized using dynamic programming. The solution to the game involves endogenous value functions, which give the pay-off to an importer as a function of the current stock, and an endogenous price function. The latter gives the price at each stage as a function of the stock size and the levels of policy variables that players use at that stage. In equilibrium, importers’ current policies are functions of the current stock, so in equilibrium the current price is a function of only the current stock. Importers are free to use nonequilibrium policies in the present, but they cannot commit themselves to the use of such policies in the future, in light of the requirement of perfection. The dependence of the current price on the current policies is of the same order of magnitude as E. This fact stems from the exporter’s ability to arbitrage, and from the Markov assumption, which means that the exporter expects future import policies to depend on future values of the state variable. If E, the length of the current period, is small, then anything an importer chooses to do in the current period (i.e. any type of nonequilibrium behaviour) has only a small effect on the trajectory of future behaviour, and can therefore have only a small effect on the current price (which depends on future behaviour). As E+O, the importers become unable to alter the current price, but this does not imply the demise of their market power. The reason is that their current actions have a first-order effect on the current rate of sales, and thus on the evolution of the state, and thus on the trajectory of future prices. The fact that, in the limit, importers cannot alter the current price does, however, have an important implication. It means that the equilibrium does not depend on whether each importer chooses a state-dependent tariff rule, or a state-dependent import quantity. In static models, or in the discrete stage dynamic model, the equilibrium depends on the choice of the policy instrument. In such models, this dependence is owing to the fact that agents face different optimization problems, and thus make different decisions, depending on what aspect of their rivals’ behaviour they take as given. For example, if importers choose import quantities, then other importers act (under the assumption of Nash behaviour) as if they are unable to affect those quantities in the current stage. If, on the other hand, agents choose tariffs, then the fact that the current price depends on the current level of policy variables (for E>O) is important. Agent i knows that his current policy will affect the current world price, and thus affect his rivals’ import levels at the current stage. Agent i’s optimization problem is consequently different when he takes as given his rivals’ tariffs rather than import quantities. The same type of argument applies if the comparison is between unit and ad
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rarqfj
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resources
valorem tariffs. As we noted above, in the limit as s--r0 the dependence of current price on current policy levels disappears; therefore in the limit there can be no dependence of the equilibrium on the choice of policy variables, at least for the reason described above. One might ask whether there is some subtler form of dependence, owing perhaps to dynamic interactions. However, future behaviour depends only on future values of the state (the Markov assumption again), so the choice of policy instruments could only affect future outcomes if it were able to affect the evolution of the state. This requires that the choice of policy instruments affect the equilibrium rate of extraction at a point in time; but, as we have just argued, for an arbitrary level of the stock, the equilibrium rate of extraction at any point in time is independent of the choice of policy instruments (in the limit as E-PO). The simplest case to analyze is one in which all n importers are identical. The price at any date is a function of the stock, P(S); aggregate equilibrium demand of importers other than i is some function y(S). For the case of linear demand [i.e. Q=fi(P--p)] and constant costs, the instantaneous level of welfare (in cash terms) of a country is [j7-P(S)-nq/2]q, where I/n is the (competitive) share of the country, and /? in the demand equation has been set to unity by choice of units. The dynamic programming equation for a representative importer is
rJ(S)=m;x{[p-P(S)-
y]q-J&(S)+q]]
where q is the importer’s rate of consumption value of S. The first-order condition is D--nq-P(S)-J,=O,
‘7
and J,=dJ/dS
(7)
is his shadow
(8)
which says that the difference between domestic and world price (the unit tariff) equals the importer’s shadow value of the resource.’ Eq. (8) has two important implications, neither of which depends on the linear functional form. When consumption is positive the importer’s consumer surplus is positive: the importer’s present discounted utility would be increased by a small increase in the stock, so the shadow value of the
‘The same first-order condition holds for the (dynamically inconsistent) open loop tariff [Karp and Newbery (1989b)], but the shadow values in the two cases obey different equations of motion; thus, the open loop tariB and the perfect tariff considered here are different.
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resource is positive. The first implication of (8), then, is that the tariff is positive on the equilibrium path, provided that consumption is positive and the stock is finite. The second implication is that the equilibrium tariff approaches zero as the stock of the resource approaches zero or becomes infinite. Therefore, for sufficiently large resource stocks the equilibrium tariff first increases and then decreases over time. To see that the tariff approaches zero as the stock approaches zero, note that the price received by the importer must approach the choke price, j, as exhaustion approaches; if this were not the case the price function would be discontinuous at S=O.* The economic interpretation is that as the stock decreases, aggregate supply (a flow) must approach zero. As the aggregate supply approaches zero, so must the excess supply facing any importer. Therefore the elasticity of excess supply becomes infinite, and it becomes optimal to impose a zero tariff. As the stock becomes infinite the scarcity value of the resource approaches zero; that is, the resource becomes identical to a standard product that can be produced at constant cost. Under these conditions the excess supply curve facing any importer is perfectly elastic, and the optimal tariff is again zero. To complete the derivation, substitute nq=Q into (8) to obtain: j5-P(S)-J,=nq=Q=
-dS/dt.
(9)
Then use nq= y+q and (9) in (7) and rearrange to obtain:
rJ=
P-p-O2n
l)Js -
(p-p-
Jd
(10)
Eq. (1) describes the response of suppliers to any deviations from the planned profile of imports, and as p= P(S), dpfdt = P,dSfdt. Replacing dS/dt by -Q, given by (9), enables us to rewrite (1) (after some rearrangement) as
J
_dP-4 S----+j-p. Ps
(11)
*There cannot be an upward jump in price at the exhaustion time, since this would violate the sellers’ arbitrage condition; if there was a downward jump in price at the exhaustion time, there would be a linal interval during which time the importers were paying more than their choke price, which would not be optimal.
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Prices 10
Tariff/rem
(log)
r
6-
Tariff/rent
(RHS)
2-
0, 0.125
,
II 0.5
1
II
I
2
I
I
8
I
I
/I
0.001
128
32
Stock (log scale) Fig. 1. Tariffs as a function
of stock (n=3,r=59,). Key: PEMF price; - - - PEMF
PlMF t/rent.
price; ..........
PIMF
t/rent;
Eqs. (10) and (11) comprise a system of ordinary differential equations in S, with boundary conditions P(0) =p and J(0) =O.j 4. Comparing the equilibrium tariffs It is relatively easy to study the comparative statics of the PEMF equilibrium tariff. The tariff is monotonic in the remaining stock, S; as S increases, the level of the tariff tends to some finite amount, A/n; as the stock approaches zero, the tariff falls to zero. The ratio of the tariff to the rent tends to infinity as the stock increases. As the rate of discount, r, increases, the ratio of the tariff to the rent increases. The comparative statics of the PIMF equilibrium tariff are more complicated, and hence we resort to numerical calculations, whose results are presented in the following graphs. Fig. 1 graphs the prices and the ratios of tariff to rent for the two -‘Eq. (10) implies that J,(O)=O, and (I I) implies that P,(O)= 30. The in&trite derivative of ps makes the system of equations ‘stilf’ and requires rather delicate techniques. The pair of ordinary differential equations were integrated numerically using the NAG Fortran Library Routine DOZEBF, using a variable-order, variable-step method of implementing the Backward Differentiation Formulas described in Hall and Watt (1976). The output was read into a PC spreadsheet program to produce the graphics, which are accurate plots.
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Tariff levels 3.5
2
8
32
128
Stock (log scale) Fig. 2. Tariff levels as discount rate varies (n=3). Key: PIMF 5%; PEMF 5%; - - - PEMF 15%.
PIMF 15%; .........
equilibria as functions of remaining stock (on a log scale), using base parameters jj= 10, c= 1, I= 5 percent, and n=3 countries.4 The ratio of the tariff to rent is read off the right-hand scale, which is logarithmic. The ratio of tariff to rent increases monotonically with stock; we showed analytically that this holds under PEMF. Fig. 2 shows the sensitivity of the results to changes in the discount rate, r. For high stock levels, an increase in the discount rate decreases the equilibrium PIMF tariff; as the stock is run down the relationship is reversed. This result illustrates that the comparative statics of the PIMF tariff are ambiguous, since they depend on the stock level. The PEMF tariff increases with the discount rate, as predicted by the analytic results reported above. The welfare properties of the two tariffs are quite different. For low stock ‘It is natural to compare the tariff with the rent, as the optimal precommitted open loop tariff has a constant ratio to the rent, and the rent represents the ‘taxable surplus’ that might reasonably be appropriated by the importers, especially if acting in concert. ‘See Bergstrom (1981) and Bergstrom, Cross and Porter (1981).
296
L. Karp and D.M. Newbery, Optimal tariffs on exhaustible resources PEMF/PIlMF 1.35
welfare
I
------------------__
l---------
I
0.95 0.125
1 0.5
I
I 2
I
I,
I 1
8
32
I
1 128
Stock (log scale) Fig. 3. Importer
welfare
as n varies (ratio of PIMF to PEMF importers; eight importers.
welfare,
r=57;).
Key: -
two
levels the importers’ welfare is higher under PEMF. ,Fig. 3 shows that importers gain approximately 9 percent under PEMF (relative to PIMF) when there are eight buyers; the gain is approximately 30 percent when there are two buyers. This comparison is quite insensitive to changes in stock levels, provided that the stock is not large. Being the ‘first-mover is a disadvantage to the importers in this situation. However, for very large stock levels, buyers do better under PIMF than under PEMF. This result can be explained by the following argument. We noted above that as the stock level approaches infinity, the buyers’ shadow value of the resource, and hence the PIMF tariff, approaches zero; the PEMF tariff, on the other hand, converges to A/n>O. In the limit as S+cc the model collapses to that of a reproducible good, produced at constant cost c. In that case the aggregate supply function is perfectly elastic and the optimal tariff equals zero, which is the equilibrium (limiting) PIMF tariff. That is, in the limiting case the PIMF tariff maximizes joint buyer welfare (as well as social surplus), and hence maximizes the welfare of importer i. The limiting PEMF tariff is positive, however. The
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assumption that exporters move first means that importers take the aggregate supply function as perfectly inelastic. and use positive tariffs in equilibrium.’ The above argument relies on a comparison of the importers’ flow of welfare in the two games, in the limit as S-+x). Numerical experiments confirm that the result also holds for large but finite stock levels. For large stock levels, a high discount rate increases the likelihood that buyers do better under PIMF, relative to PEMF. The reason is that for large stock levels the games ‘are close to’ the limiting games described above. and in those games the flow of welfare is higher under PIMF than PEMF: and for high discount rates the present value of the stream of welfare is dominated by the flow near the beginning of the program (while the stock is large). The previous comments imply that if the stock levels and the discount rate are sufhciently large, the PEMF tariff is disadvantageous.6 This means that importers would have a higher present value of welfare if they could commit to using a zero tariff throughout the program. The possibility of disadvantageous market power was noted by Maskin and Newbery (1978, revised 1990) in the context of a two-period nonrenewable resource model. That model assumed a single dominant importer facing competitive producers and competing buyers; all buyers behave strategically in our model. Maskin and Newbery study the situation where the dominant importer moves first in each period, so the natural comparison is with the PIMF tariff. Owing to the strategic symmetry of buyers in our model, however, we do not find any situation where the PIMF tariff is disadvantageous.
5. Conclusions
We have derived and characterized the Markov perfect equilibria for two games in which dominant buyers use tariffs to exert market power when confronted by competitive producers. The two games embody different assumptions regarding the ability of producers to arbitrage supply across time. In neither game do producers have any incentive to change their behaviour in equilibrium. If importers move first .- the PIMF assumption exporters would be able to change the level of aggregate exports in response ‘This behaviour does not reflect irrationality or myopia on the part of any agent; it is simply a consequence of the Markov assumption and of the order of play. The Markov assumption means that current buyers’ behaviour cannot affect the sellers’ future decisions. Therefore, the buyers cannot do better than to maximize their welfare in the current period; Karp (1987) obtains a similar result for the case of a single buyer. 6For large stock levels, when the supply is nearly perfectly elastic, a zero tariff results in a higher flow of welfare to importers than does a tariff that is close to A/n. For high discount rates, any comparison of payoffs under dilferent equilibria depends on the comparison of the flow of welfare near the beginning of the trajectory.
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to a deviation from equilibrium on the part of importers; they lack this ability if they must move first - the PEMF assumption. The equilibrium PEMF tariff is a monotonic function of time, but for large initial stocks the PIMF tariff first increases then decreases. The PEMF tariff is greater than the PIMF tariff, for given stock levels. Provided that the initial stock is not too large, importers do better under PEMF; that is, there is a first-mover disadvantage. This is reversed for very large stock levels. If the stock level is sufficiently large, market power is disadvantageous under the PEMF assumption. In addition to the specific results described below, this paper illustrates techniques that are useful in studying imperfectly competitive resource markets. The most important advantage of these methods is that they permit the study of perfect equilibria; previous papers have concentrated on open loop equilibria or used two-period models. There are a number of promising directions for future research. It would be useful to generalize the model to the case of asymmetric buyers, in order to study the situation where there are both competitive and oligopsonistic buyers in the market. This would be straightforward. It would also be interesting to’ incorporate noncompetitive sellers. This is a more challenging problem, since it would require a multi-dimensional state vector, consisting of the stock levels of all sellers.
References Bergstrom, Theodore, 1981, On capturing oil rents with national excise tax, American Economic Review 71, 194201. Bergstrom, Theodore, John Cross and Richard Porter, 1981, Efiiciency inducing taxation for a monopolistically supplied depletable resource, Journal of Public Economics 15, no. 1, 23-32. Hall, G. and J.M. Watt, eds., 1976, Modern numerical methods for ordinary differential equations (Clarendon Press, Oxford). Karp, Larry, 1987, Consistent tariffs with dynamic supply response, Journal of International Economics 23, 369-376. Karp, Larry and David Newbery, 1989a, Intertemporal consistency issues in depletable resources, Discussion Paper no. 346 (Centre for Economic Policy Research, London), in: James Sweeney and Allen Kneese, eds., Handbook of natural resource and energy economics, Vol III (North-Holland, Amsterdam), forthcoming. Karp, Larry and David Newbery, 1989b, Time consistent oil import tariffs, Discussion Paper no. 344 (Centre for Economic Policy Research, London). Karp, Larry and David Newbery, 1989~. Optimal tariffs on exhaustible resources, Economic Theory Discussion Paper no. 141 (Department of Applied Economics, Cambridge, England). Kemp, Murray and Ngo Long, 1980, Optimal tariffs and exhaustible resources, in: Murray Kemp and Ngo Long, eds., Exhaustible resources, optimality and trade, Essay 18 (NorthHolland, Amsterdam). Maskin, Eric and David Newbery, 1978, Rational expectations with market power: The paradox of the disadvantageous monopolist, Warwick Economic Discussion Paper and Massachusetts Institute of Economics Discussion Paper (Cambridge, MA). Maskin, Eric and David Newbery, 1990, Disadvantageous oil tariffs and dynamic consistency, American Economic Review 80, no. 1, 143-156.
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Newbery,
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resources
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Newbery. David, 1976, A paradox in tax theory: Optimal tarilTs on exhaustible resources, SEER Technical Paper (Stanford Economics Department). Olsen, Trond, 1989, A folk theorem for rent extracting tariffs on exhaustible resources. Unpublished working paper (Norwegian Center for Research in Organization and Management, Bergen. Norway). Reinganum. Jennifer and Nancy Stokey, 1985, Oligopoly extraction of a common property natural resource: The importance of the period of commitment in dynamic games, International Economic Review 26 (I), 161-173.