S .J. Rubio, L. Escriche Journal of Public Economics 79 2001 297 –313
299
which importing countries are leader. Section 6 summarizes the conclusions and suggests directions for future research.
2. The model
We use a slightly modified version of the one-state variable model presented in Wirl 1995. Assuming that the consumers of the importing countries act as
price-taker agents, we can write the discounted present value of the consumers’ net
` 2
d t 2
2
welfare as: e e
haqt 2 1 2qt 2 [ pt 1 ct]qt 1 Rt 2 ´zt j dt, where
2
d is the discount rate, aqt 2 1 2qt the consumers’ gross surplus, qt the amount of the resource bought by the importing countries, pt the producer price,
ct the tax fixed by the importing country governments, Rt a lump-sum transfer
2
that the consumers receive from the government, and ´zt the environmental or
pollution damage function, where zt denotes the cumulative emissions and ´
stands for a positive parameter. Global warming is a clear example of a stock externality
. Hence, consumers take as given not only the price of the resource, but also the evolution of the accumulated emissions and the lump-sum transfer. The
resource demand thus depends only on the consumer price: qt 5 a 2 pt 2 ct.
The governments of the importing countries tax emissions in order to maximize the discounted present value of the net consumers’ surplus. They reimburse tax
revenues as lump-sum transfers to consumers, so that the after-tax consumers’ net
2
welfare does not depend on tax revenues. The optimal time path for the tax is thus given by the solution of the following problem:
`
1
2 d t
2
]
H
max
E
e aa 2 pt 2
ct 2 a 2 pt 2
ct 2 pta 2 pt 2
hc t j 2
J
2 ct 2 ´zt
dt. 1
The dynamics of cumulative resource consumption determine simultaneously the dynamics of the CO concentration in the atmosphere. We are assuming that
2 3
depreciation of cumulative emissions is zero so that emissions are irreversible: ~
zt 5 a 2 pt 2 ct, z0 5 z 0.
2 Let us turn to the other side of the market. We assume that extraction costs
2
This is the main difference with Wirl’s model. In this respect we follow Tahvonen’s 1996 approach.
3
This simplified version of the cumulative emission dynamics, where cumulative extractions are used as a proxy of CO concentration, has also been employed by Hoel 1992, 1993, Wirl 1994, Wirl and
2
Dockner 1995, and Tahvonen 1995, 1996.
300 S
.J. Rubio, L. Escriche Journal of Public Economics 79 2001 297 –313
depend linearly on the rate of extraction and on the cumulative extraction. The objective of the cartel of producers is to define a price strategy that maximizes the
4
present value of profits:
` 2
d t
max
E
e pt 2 czta 2 pt 2
ct dt. 3
h j
h pt j
Tahvonen 1995, 1996 considered the possibility of a non-linear function for extraction costs; the assumption of linear costs with respect to the extraction rate,
however, has been extensively used in the economic literature of non-renewable resources — see, e.g., Ulph and Ulph 1994, Wirl 1995, Farzin 1996, Farzin
and Tahvonen 1996 and Hoel and Kverndokk 1996. If we assume that average extraction costs are increasing with respect to the extraction rate, we are implicitly
assuming that the extraction technology exhibits decreasing returns to scale; but decreasing returns must reflect the scarcity of an underlying, unlisted factor of
production distinct from the natural resource. For this reason, in a long-run analysis when all production factors are considered variable, the constant-returns
assumption is more reasonable than the decreasing-returns assumption.
We also assume the stock externality to be largely irrelevant to the welfare of exporting countries, and the cumulative extractions to be constrained not by the
resource in the ground, but by its negative impact on extraction costs and climate. Moreover, we assume that the producers get no utility from consuming the
5
resource. We represent the strategic interactions in the resource market as a differential game between a coalition of importing-country governments and a
cartel of oil exporters, where the coalition of importing countries chooses the tax and the cartel the price.
3. A neutral pigouvian tax
