Abstract
The paper presents a review of mathematical models used in economic analysis of environmental problems. This area of research combines macroeconomic models of growth, as dependent on capital, labour, resources, etc., with environmental models describing such phenomena like natural resources exhaustion or pollution accumulation and degradation. In simpler cases the models can be treated analytically and the utility function can be optimized using, e.g., such tools as the maximum principle. In more complicated cases calculation of the optimal environmental policies requires a computer solution.
Similar content being viewed by others
References
T.J. Allen, Managing the Flow of Technology. Technology Transfer and the Dissemination of Technological Information within R & D Organization (MIT Press, Cambridge, 1977).
K.J. Arrow and M. Kurtz, Public Investment, the Rate of Return, and Optimal Fiscal Policy(John Hopkins Press, Baltimore, 1970).
J. Barro, Goverment spending in a simple model of endogenous growth, J. Political Economy 98 (1990) 103–125.
R.J. Barro and X. Sala-i-Martin, Economic Growth(McGraw-Hill, New York, 1995).
T. Basar and G. Olsder, Dynamic Noncooperrative Game Theory(Academic Press, New York, 1982).
W.J. Baumol and W.E. Oates, The Theory of Environmental Policy(Cambridge University Press, Cambridge, MA, 1988).
G.S. Becker, K.M. Murphy and R. Tamura, Human capital, fertility, and economic growth, J. Political Economy 98(5) (1990) 12–37.
K.E. Boulding, The economics of the coming spaceship earth, in: Environmental Quality in a Growing Economy, ed. H. Jarret (John Hopkins Press, Baltimore, 1966) pp. 3–14.
A.L. Bovenberg and R.A. de Mooij, Environmental levies and distortionary taxation, Amer. Econ. Review 84(4) (1994) 1085–1089.
G. Chichilnisky, Economic development and efficiency criteria in the satisfaction of basic needs, Appl. Math. Modeling 1(6) (1977) 290–297.
G. Chichilnisky, An axiomatic approach to sustainable development, Social Choice and Welfare 13 (1996) 231–257. Z. Nahorski, H.F. Ravn / Mathematical models in economic environmental problems199
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources(Wiley, New York, 1976).
R.H. Coase, The problem of social cost, J. Law Econ. 3 (1960) 1–44.
J.M. Conrad and C.W. Clark, Natural Resource Economics(Cambridge University Press, New York, 1987).
T.D. Crocker, Structuring of atmosferic pollution control systems, in: The Economics of Air Pollution, ed. H. Wolozin (Norton, New York, 1966).
M.L. Cropper and W.E. Oates, Environmental economics: a survey, Journal of Economic Literature 30 (1992) 675–740.
R.C. D'Arge and K.C. Kogiku, Economic growth and the environment, Review of Economic Studies 40 (1973) 61–77.
P.S. Dasgupta, The Control of Resources(Basil Blackwell, Oxford, 1982).
P.S. Dasgupta and G.M. Heal, The optimal depletion of exhaustible resources, in: Review of Economic Studies (1974) pp. 3–28.
P.S. Dasgupta and G.M. Heal, Natural Resource Economics(Cambridge University Press, Cambridge, 1979).
R. Davison, Optimal depletion of an exhaustible resource with research and development towards an alternative technology, Review of Economic Studies 45 (1978) 355–367.
P.B. Dixon, B.R. Parmenter, J. Sutton and D.P. Vincent, Notes and Problems in Applied General Equilibrium Economics (North-Holland, 1992).
C.C. Duarte, Renewable resource market obeying difference equations: stable points, stable cycles, and chaos, Environmental and Resource Economics 4 (1994) 353–381.
G.C. Evans, The dynamics of monopoly, American Math. Monthly 31 (1924) 77–83.
G. Feichtinger and R.F. Hartl, Optimale Kontrolle ¨okonomischer Prozesse(De Gruyter, Berlin, 1986).
A.C. Fischer and F.M. Peterson, The environment in economics: a survey, Economic Literature 14 (1976) 1–31.
H. Folmer and C. Howe, Environmental problems and policy in the single european market, Environmental and Resource Economics 1(1) (1991) 17–41.
H. Folmer and I. Musu, eds., Environmental cooperation and policy in the single European market, Environmental and Resource Economics 2(2) (1992).
B.A. Forster, A note on the optimal control of pollution, Journal of Economic Theory 5 (1972) 537–539.
B.A. Forster, On a one state variable optimal control problem-consumption-pollution trade-off, in: Applications of Control Theory to Economic Analysis, eds. J.D. Pitchford and S.J. Turnovsky (North-Holland, 1977) pp. 35–56.
V. Ginsburgh and M. Keyzer, Structure of Applied General Equilibrium Models(MIT Press, 1997).
B. Gompertz, On the nature of the function expressive of the law of human mortality, Philos. Trans. R. Soc. 115 (1825) 513–585.
H. Gordon, Economic theory of a common-property resource: the fishery, Journal of Political Economy 62 (1954) 124–142.
H. Gordon, A reinterpretation of the pure theory of exhaustion, Journal of Political Economy 75 (1967) 274–286.
D. Gottwald, Die dynamische Theorie der Allokation ersch¨opfbarer Ressourcen(Vandenhoek and Ruprecht, G¨ottingen, 1981).
L.H. Goulder, Environmental taxation in a second-best world, in: The International Yearbook of Environmental and Resource Economics 1997/1998, eds. H. Folmer and T. Tietenberg (Edward Elgar, Chelthenham, 1997) pp. 28–54.
H. Halkin, Optimal control for systems described by difference equations, in: Advances in Control Systems, Vol. I, ed. C.T. Leondes (Academic Press, 1964) pp. 173–196.
H. Hotelling, The economics of exhaustible resources, Journal of Political Economy 39 (1931) 137–175.
S.E. Jørgensen, Fundamentals of Ecological Modelling(Elsevier, Amsterdam, 1986).
M.I. Kamien and N.L. Schwartz, Optimal exhaustible resource depletion with endogenous technical changes, Review of Economic Studies 45 (1978) 179–196.
M.I. Kamien and N.L. Schwartz, Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management (North-Holland, New York, 1981).
E. Keeler, M. Spence and R. Zeckhauser, The optimal control of pollution, J. Economic Theory 4 (1972) 19–34.
N. Keyfitz, Introduction to Mathematics of Population(Addison-Wesley, Reading, MA, 1968).
N. Keyfitz, Applied Mathematical Demography(Springer, New York, 1985).
J.V. Krutilla, Conservation reconsidered, American Economic Review 57 (1967) 777–786.
G. Leitmann, Cooperative and Non-Cooperative Many Players Differential Games(Springer, Vienna, 1974).
A.J. Lotka, Elements of Physical Biology(Williams and Wilkins, Baltimore, 1925).
R.E. Lucas, On the mechanics of economic development, J. Monetary Development 22 (1988) 3–42.
M. Luptacik and U. Schubert, Optimale investitionspolitik unter ber¨ucksichtigung der umwelt: Eine anwendung der kontrolltheorie, in: Operations Research Verfahren, Vol. 35, eds. W. Orttli and F. Steffens (Athen¨aum/Hain, Meisenheim, 1979) pp. 271–282.
M. Luptacik and U. Schubert, Optimal investment policy in productive capacity and pollution abatement processes in a growing economy, in: Optimal Control Theory and Economic Analysis, ed. G. Feichtinger (North-Holland, Amsterdam, 1982) pp. 231–243.
K.G. M¨aler, Environmental Economics(John Hopkins Press, Baltimore, 1974).
K.G. M¨aler, The acid rain game, in: Valuation Methods and Policy Making in Environmental Economics, eds. H. Folmer and E. van Ierland (Elsevier, Amsterdam, 1989) pp. 221–252.
K.G. M¨aler, International environmental problems, Oxford Review of Economic Policy 6 (1990) 80–108.
T.R. Malthus, Principle of Population as it affects the Future Improvement of Society, with Remarks on the Speculations of Mr. Godwin, M. Condorcet and other Writers. Reprinted as: Population: The first Essay (The University of Michigan Press, Ann Arbor, 1959). Original edition 1798.
A.S. Manne and R.G. Richels, Buying Greenhouse Insurance(MIT Press, Cambridge, MA, 1992).
A.P. Martins, Human capital and economic growth, Ph.D. thesis, Columbia University (1987).
R.M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theor. Biol. 51 (1975) 511–524.
D.H. Meadows, D.L. Meadows, J. Randers and W.W. Behrens III, The Limit to Growth(Universe Books, New York, 1972).
A. Mehlmann, Applied Differential Games(Plenum Press, New York, 1988).
M. Mesarovic and E. Pestel, Mankind at the Turning Point(Dutton, New York, 1974).
J.S. Mill, Principles of Political Economy(Augustus M. Kelly, New York) reprint of sixth edition, 1965 edition (1848).
S.R. Milliman and R. Prince, Firm incentives to promote technological change in polution control,Journal of Environmental Economics and Management 77 (1989) 247–265.
J. Monod, Recherches sur la Crossance de Culture Bact´eriennes(Hermann, Paris, 1942).
J. Monod, La technique de culture continue: theorie et applications, Annals Institute Pasteur 79 (1950) 390–410.
W.D. Montgomery, Markets in licenses and efficient pollution control programs, Journal of Economic Theory 5 (1972) 395–418.
W.D. Nordhaus, Managing the Global Commons. The Economics of Climate Change(MIT Press, Oxford, MA, 1994).
E.C. Pielou, Mathematical Ecology(Wiley, New York, 1977).
A.C. Pigou, The Economics of Welfare(Macmillan, London, 1932). Z. Nahorski, H.F. Ravn / Mathematical models in economic environmental problems 201
F. van der Ploeg and C. Withagen, Pollution control and the Ramsey problem, Environmental and Resource Economics 1 (1991) 97–118.
F. van der Ploeg and A.J. de Zeeuw, International aspects of pollution control, Environmental and Resource Economics 2(2) (1992) 117–139.
C. Plourde, A model of waste accumulation and disposal, Canadian J. Econ. 5 (1972) 119–125.
F.P. Ramsey, A mathematical theory of saving, Economic J. 38 (1928) 543–559.
J. Rawls, A Theory of Justice(Bellknap Press, Cambridge, MA, 1971).
S. Rebelo, Long run policy analysis and long run growth, J. Political Economy 99 (1991) 500–521.
E.G. Roberts, The Dynamics of Research & Development(Harper & Row, 1964).
P.M. Romer, Increasing returns and long run growth, J. Political Economy 94 (1986) 1002–1037.
M.B. Schaefer, Some considerations of population dynamics and economics in relation to the management of marine fisheries,J. Fisheries Res. Board of Canada 14 (1957) 669–681.
W.D. Schultze and R.C. D'Arge, The Coase proposition, information constraints, and long-run equilibrium, American Economic Review 64 (1974) 763–772.
A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications(North-Holland, Amsterdam, 1987).
J. Showen and J. Whalley, Applied general-equilibrium models of taxation and international trade: an introduction and survey, Journal of Economic Literature 22 (1984) 1007–1051.
J. Showen and J. Whalley, Applying General Equilibrium(Cambridge University Press, 1992).
H. Siebert, ¨ Okonomische Theorie nat¨urlicher Ressourcen(Mohr, T¨ubingen, 1983).
H. Siebert, Economics of the Environment(Springer, Heidelberg, 1998).
V.L. Smith, Dynamic of waste accumulation: disposal versus recycling, Quart. J. of Econ. 76 (1972) 600–616.
V.L. Smith, Control theory applied to natural and environmental resources, Journal of Environmental Economics and Management 4 (1977) 1–24.
R.M. Solow, Intergenerational equity and exhaustible resources, in: Review of Economic Studies(1974) pp. 29–45.
J. Stiglitz, Growth with exhaustible resources: efficient and optimal growth paths, in: Review of Economic Studies (1974) pp. 123–137.
J. Stiglitz, Growth with exhaustible resources: the competitive economy, in: Review of Economic Studies (1974) pp. 139–152.
O. Tahvonen, On the dynamics of renewable resource harvesting and pollution control, Environmental and Resource Economics 1 (1991) 97–117.
O. Tahvonen, Dynamics of pollution control when damage is sensitive to the rate of pollution accumulation, Environmental and Resource Economics 5 (1995) 9–27.
R. Tamura, Income convergence in an endogenous growth model, J. Political Economy 99(3) (1991) 522–540.
P.F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance Math´ematique et Physique 10 (1838) 113–121.
V. Volterra, Lec¸ons sur la Th´eorie Math´ematique de la Lutte pour la Vie(Gauthier-Villars, Paris, 1931).
M. Weinstein and R. Zeckhauser, Use patterns for depletable and recyclable resources, in: Review of Economic Studies (1974) pp. 67–88.
von Weizacker, Lemmas for a theory of approximate optimal growth, Rev. Econ. Studies 34 (1967) 143–151.
C. Withagen, Pollution, abatement and balanced growth, Environmental and Resource Economics 5 (1995) 1–8.
A. Xepapadeas, Advanced Principles in Environmental Policy(Edward Edgar, Cheltenham, 1997).
T. Żylicz, Improving the environment through permit trading: the limits to a market approach, in: International Environmental Economics Theories, Models and Applications to Climate Change, International Trade and Acidification, ed. E.C. van Ierland (Elsevier, 1994) pp. 283–306.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nahorski, Z., Ravn, H.F. A review of mathematical models in economic environmental problems. Annals of Operations Research 97, 165–201 (2000). https://doi.org/10.1023/A:1018913316076
Issue Date:
DOI: https://doi.org/10.1023/A:1018913316076