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Market Power, Resource Extraction and Pollution: Some Paradoxes and a Unified View

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Green Growth and Sustainable Development

Part of the book series: Dynamic Modeling and Econometrics in Economics and Finance ((DMEF,volume 14))

Abstract

We adopt a stepwise approach to the analysis of a dynamic oligopoly game in which production makes use of a natural resource and pollutes the environment, starting with simple models where firms’ output is not a function of the natural resource to end up with a full-fledged model in which (i) the resource is explicitly considered as an input of production and (ii) the natural resource and pollution interact via the respective state equations. This allows us to show that the relationship between the welfare properties of the economic system and the intensity of competition is sensitive to the degree of accuracy with which the model is constructed.

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Notes

  1. 1.

    This view is so largely shared in the profession, that it appears regularly in the introductory chapters of textbook at any level (see, e.g., Pearce and Turner 1989; Tisdell 2009; and Anderson 2010).

  2. 2.

    See Downing and White (1986), Milliman and Prince (1989), Karp and Livernois (1994), Chiou and Hu (2001) and Poyago-Theotoky (2007), inter alia.

  3. 3.

    Classical contributions in this vein are those of Clark and Munro (1975) and Levhari and Mirman (1980). For a model of international trade with natural resource extraction, see Copeland and Taylor (2009), inter alia. An overview of the debate is in Long (2010).

  4. 4.

    See, for instance, Chap. 12 in Dockner et al. (2000) and Chaps. 2–3 in Long (2010), offering surveys of dynamic games in which environmental externalities and resource extraction are treated separately.

  5. 5.

    We could model the state equation as

    $$ \dot{x} ( t ) =\eta x ( t ) -\nu \sum_{i=1}^{n}q_{i} ( t ) $$

    with ν∈(0,1]. This, however, would not modify significantly the qualitative predictions of the our analysis. Therefore, we have imposed ν=1 to restrict the set of parameters.

  6. 6.

    Taking x 0>0 as the initial condition does not ensure the sustainability of extraction activities over t∈[0,∞) as the stock x(t) would become nil in finite time.

  7. 7.

    Henceforth, we omit the time argument for the sake of brevity. Mangasarian’s (1966) and Arrow’s (1968) sufficiency conditions are also satisfied. They are also omitted for the sake of brevity.

  8. 8.

    Throughout the paper, we shall use superscript N to identify the Nash solution, while starred values will indicate steady state magnitudes of states and controls. Note that in the present section the optimal output is stationary as firms do not take into account the consequences of their behaviour on the stock of resources. Hence, here the steady state output is the same as the Nash equilibrium output at any time t. The same applies to the model illustrated in Sect. 2.3.

  9. 9.

    This is the reason why we have taken the initial stock to be at least as large as the Cournot-Nash industry output.

  10. 10.

    Note that the amount of natural resource enters the social welfare function with a weight equal to one, i.e., the same attached to industry profits and consumer surplus. The ongoing debate on this point has not yet produced a unanymous view (see, e.g., Chap. 5 in Stern 2009). The need for guaranteeing the prosperity of future generations suggests that one should attach to the preservation of natural resources at least the same importance as traditional economic indicators strictly related to production and consumption.

  11. 11.

    Additionally, note that it is not defined in linear-quadratic form. Consequently, we have no obvious conjecture as to the form of the value function.

  12. 12.

    The corresponding value of the co-state variable at the steady state equilibrium is

    $$ \lambda^{\ast}=\frac{ ( a-c ) ( n-1 ) \eta}{n [ 2\eta- ( n+1 ) \rho ] } $$

    and the transversality condition lim t→∞ e −ρt λ i x=0 is met thanks to exponential discounting.

  13. 13.

    This result relates the long-run effects of increasing the population of firms with the discount rate. This aspect is a crucial feature of an ongoing debate concerning the need for applying low discount rates to the well-being of future generation (see Stern 2007; and Weitzman 2007, inter alia).

  14. 14.

    Note that μ i =0 suffices to ensure that the transversality condition lim t→∞ μ i s=0 be satisfied.

  15. 15.

    Therefore, we have a single state, s and 2n+1 controls, i.e., q=(q 1,q 2,…,q n ) and k=(k 1,k 2,…,k n ), two for each firm, and the Pigouvian policy rate θ for the government.

  16. 16.

    This is the route taken by Benchekroun and Long (1998, 2002).

  17. 17.

    It is worth noting that this mechanism would still exist in a simpler version of this setup, without R&D investments. This is due to the fact that the conflict between two equally desirable objectives (lowering the price and reducing pollution) is entirely inherent in production decisions only.

  18. 18.

    The proof, trivial but lengthy, is omitted for brevity. It is however available from the authors upon request.

  19. 19.

    This applies to rain forests and the oceans absorbing CO2 emissions, while it does not apply to other natural resources, like the stock of fish. On the contrary, the latter is negatively affected by pollution (as specified in (2), (10) and (42)).

  20. 20.

    This result has a definite Arrowvian flavour. For a summary of the debate between Schumpeter (1942) and Arrow (1962) on the relationship between market power and innovation incentives, see, e.g., Tirole (1988).

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Acknowledgements

We would like to thank Roberto Cellini, Arsen Palestini, Tapio Palokangas (Editor), an anonymous referee and the audience at the workshop Green Growth and Sustainable Development (IIASA, Laxenburg, 9–10 December 2011) for useful comments and suggestions. The usual disclaimer applies.

Author information

Authors and Affiliations

  1. Department of Economics, University of Bologna, Strada Maggiore 45, Bologna, 40125, Italy

    Luca Lambertini

  2. College of Engineering, University of California at Berkeley, Berkeley, CA, 94720, USA

    George Leitmann

Authors
  1. Luca Lambertini
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Corresponding author

Correspondence to Luca Lambertini .

Editor information

Editors and Affiliations

  1. and Business (WU), Department of Economics, Vienna University of Economics, Augasse 2-6, Vienna, 1090, Austria

    Jesús Crespo Cuaresma

  2. Political and Economic Sciences, University of Helsinki, Arkadiankatu 7, Helsinki, 00014, Finland

    Tapio Palokangas

  3. UrB RAS, ILASA, Dynamic Systems, Institute of Mathematics and Mechanics, S. Kovalevskaya 16, Ekaterinburg, 620990, The Sverdlovsk Area, Russia

    Alexander Tarasyev

Appendix: List of Symbols

Appendix: List of Symbols

Parameters, variables and functions appearing in text are defined as follows:

a::

reservation price

b i (t)::

extraction rate of firm i

\(B\equiv\sum_{i=1}^{n}b_{i}\)::

industry extraction rate

c::

marginal production cost

CS::

consumer surplus

\(\mathcal{H}_{i} ( t ) \)::

Hamiltonian function of firm i

J::

Jacobian matrix

k i (t)::

green R&D effort of firm i

n::

number of firms

p(t)::

market price

\(\mathcal{P}\)::

Pigouvian taxation

q i (t)::

quantity of firm i

Q(t)::

industry output

s(t)::

pollution stock

SW::

social welfare

v::

marginal cost of R&D

x(t)::

resource stock

z::

marginal environmental damage

δ::

natural rate of emission absorption

γ i (t), ϖ i (t), ζ i (t), ϰ i (t)::

co-state variables

η::

rate of reproduction of natural resources

θ::

Pigouvian tax rate

λ i (t), μ i (t), φ i (t), ψ i (t)::

co-state variables (in current value)

π i (t)::

profits of firm i

ρ::

discount rate

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Lambertini, L., Leitmann, G. (2013). Market Power, Resource Extraction and Pollution: Some Paradoxes and a Unified View. In: Crespo Cuaresma, J., Palokangas, T., Tarasyev, A. (eds) Green Growth and Sustainable Development. Dynamic Modeling and Econometrics in Economics and Finance, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34354-4_7

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