Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter April 9, 2019

Pollution, carrying capacity and the Allee effect

  • Stefano Bosi EMAIL logo and David Desmarchelier


In ecology, one of the simplest representation of population dynamics is the logistic equation. This basic view can be enriched by considering two important variables: (1) the maximal population density Nature can support (carrying capacity) and (2) the critical density threshold under which the population disappears (Allee effect). The economic literature on biodiversity and renewable resources ignores both these variables. Evidence suggests also that these variables are affected by the pollution level due to economic activity. Indeed, a degraded environment is unsuitable for wildlife and reduces the carrying capacity, while the climate change entails the habitat fragmentation and, lowering the wildlife reproduction possibilities, raises the Allee effect. The present paper aims to incorporate both endogenous carrying capacity and Allee effect in a Ramsey model augmented with biodiversity as a renewable resource. Our extended framework enables us to study the effect of a Pigouvian tax on anthropogenic mass extinction. We find that, when the household overvalues biodiversity with respect to consumption, a higher green-tax rate is beneficial in three respects entailing: (1) a lower pollution and a higher biodiversity, (2) a welfare improvement and (3) a less likely mass extinction.

JEL Classification: E32; O44


Proof of Proposition 1

We apply the Pontryagin’s maximum principle. The agent maximizes the intertemporal utility functional under the budget constraint (2). Setting the Hamiltonian H=eθtu(c,N)+λ[(rδ)h+wc], deriving the first-order conditions H/c=0, H/h=λ˙ and H/λ=h˙, and defining μλeθt, we get (4), (5) and (6). Notice that the second-order Arrow-Mangasarian conditions are satisfied because Nature enters the utility function as an externality.   □

Proof of Proposition 2.

Consider (4), (5), (6) jointly with (1), (9) and (12).   □

Proof of Proposition 4.

Simply differentiate (17), (18), (19) and (20).   □

Proof of Proposition 5.

Apply Proposition 4 to expression (22). We have Wi(τ)>0 if and only if


If i = 1, then the RHS is negative: the right inequality in (28) is violated and, therefore, W1(τ)<0. If i = 2, then the RHS is positive: the right inequality in (28) is satisfied if and only if (23) holds.   □

Proof of Lemma 1.

The Jacobian matrix is given by




or, equivalently, (24), while ψ is given by (25) because, around N1 or N2, the steady states we are interested in,


We observe that


because N1=A(P) and N2=C(P), and that, according to (8),


Thus, we obtain



Proof of Proposition 6.

We observe that the system (13)–(16) has one jump variable (μ) and three predetermined variables (k, N and P). In this case, local indeterminacy arises if and only if the four eigenvalues of J are stable implying D > 0 as a necessary (but not sufficient) condition for local indeterminacy. However, C(P)>A(P) entails φ2 < 0, that is D < 0.   □

Proof of Proposition 7.

If C(P)>A(P), then φ1>0 and D > 0. Three cases are possible: (1) all eigenvalues are stable (local indeterminacy), (2) all eigenvalues are unstable or (3) two eigenvalues are stable while the other two are unstable. a<θ implies T > 0 and rules out the first case (local indeterminacy).   □

Proof of Proposition 8.

Consider functions (10). The two steady state coalesce when N = N1 = N2, that is when a=a, holding if and only if φ=φ1=φ2=D=0.   □

Proof of Proposition 9.

According to Bosi and Desmarchelier (2019), a Hopf bifurcation generically arises in a 4D-system if and only if S2=S3/T+DT/S3 with S3/T>0.

S2=S3/T+DT/S3 if and only if


that is if and only if η=η±. Moreover,


Since N = N2 with C(P)>A(P), we get D < 0. Then, S3(η)/T<0<S3(η+)/T. Therefore, only η+ satisfies the inequality required for a Hopf bifurcation.   □


Allee, W. C. 1931. Animal Aggregations: A Study in General Sociology. Chicago: University of Chicago Press.10.5962/bhl.title.7313Search in Google Scholar

Ayong Le Kama, A. 2001. “Sustainable Growth, Renewable Resources and Pollution.” Journal of Economic Dynamics & Control 25: 1911–1918.10.1016/S0165-1889(00)00007-5Search in Google Scholar

Barnosky, A. D., N. Matzke, S. Tomiya, G. O. Wogan, B. Swartz., T. B. Quental, C. Marshall, J. L. McGuire, E. L. Lindsey, K. C. Maguire, B. Mersey, and E. A. Ferrer. 2011. “Has the Earth’s Sixth Mass Extinction Already Arrived?” Nature 471: 51–57.10.1038/nature09678Search in Google Scholar PubMed

Beltratti, A., G. Chichilnisky, and G. Heal. 1994. “Sustainable Growth and the Green Golden Rule.” In The Economics of Sustainable Development, edited by I. Goldin and L. A. Winters, Cambridge: Cambridge University Press.10.3386/w4430Search in Google Scholar

Bosi, S., and D. Desmarchelier. 2017. “Are the Laffer Curve and the Green Paradox Mutually Exclusive?” Journal of Public Economic Theory 19: 937–956.10.1111/jpet.12242Search in Google Scholar

Bosi, S., and D. Desmarchelier. 2018a. “Limit Cycles Under a Negative Effect of Pollution on Consumption Demand: The Role of an Environmental Kuznets Curve. Environmental and Resource Economics 69: 343–363.10.1007/s10640-016-0082-7Search in Google Scholar

Bosi, S., and D. Desmarchelier. 2018b. “Natural Cycles and Pollution.” Mathematical Social Sciences 96: 10–20.10.1016/j.mathsocsci.2018.08.005Search in Google Scholar

Bosi, S., and D. Desmarchelier. 2019. “Local Bifurcations of Three and Four-Dimensional Systems: A Tractable Characterization with Economic Applications.” Mathematical Social Sciences 97: 38–50.10.1016/j.mathsocsci.2018.11.001Search in Google Scholar

Brook, B. W., N. S. Sodhi, and C. J. A. Bradshaw. 2008. “Synergies Among Extinction Drivers Under Global Change.” Trends in Ecology and Evolution 23: 453–460.10.1016/j.tree.2008.03.011Search in Google Scholar PubMed

Ceballos, G., P. R. Ehrlich., A. D. Barnosky, A. García, R. M. Pringle, and T. M. Palmer. 2015. “Accelerated Modern Human-Induced Species Losses: Entering the Sixth Mass Extinction.” Science Advances 1: 1–5.10.1126/sciadv.1400253Search in Google Scholar PubMed PubMed Central

Celli, G., and B. Maccagnani. 2003. “Honey bees as Bioindicators of Environmental Pollution.” Bulletin of Insectology 56: 137–139.Search in Google Scholar

Courchamp, F., T. Clutton-Brock, and B. Grenfell. 1999. “Inverse Density Dependence and the Allee Effect.” Trends in Ecology and Evolution 14: 405–410.10.1016/S0169-5347(99)01683-3Search in Google Scholar

Dinda, S. 2004. “Environmental Kuznets curve Hypothesis: A Survey.” Ecological Economics 49: 431–455.10.1016/j.ecolecon.2004.02.011Search in Google Scholar

Gerlagh, R., and M. Liski. 2011. “Strategic Resource Dependence.” Journal of Economic Theory 146: 699–727.10.1016/j.jet.2010.09.007Search in Google Scholar

Hoel, M. 2010. “Climate Change and Carbon Tax Expectations.” CESIFO Working Paper n. 2966.10.2139/ssrn.1564032Search in Google Scholar

Jensen, S., K. Mohlin, K. Pittel, and T. Sterner. 2015. An Introduction to the Green Paradox: The Unintended Consequences of Climate Policies. Review of Environmental Economics and Policy 9: 246–265.10.1093/reep/rev010Search in Google Scholar

Memmott, J., N. M. Waser, and M. V. Price. 2004. “Tolerance of Pollination Networks to Species Extinctions.” Proceeding of the Royal Society B 271: 2605–2611.10.1098/rspb.2004.2909Search in Google Scholar

OECD. 2017. OECD Environmental Performance Reviews: Canada 2017. Paris: OECD Publishing.Search in Google Scholar

Sinn, H.-W. 2008. “Public Policies Against Global Warming: A Supply Side Approach.” International Tax and Public Finance 15: 360–394.10.1007/s10797-008-9082-zSearch in Google Scholar

Van der Meijden, G., F. Van der Ploeg, and C. Withagen. 2015. “International Capital Markets, Oil Producers and the Green Paradox.” European Economic Review 76: 275–297.10.1016/j.euroecorev.2015.03.004Search in Google Scholar

Wirl, F. 2004. “Sustainable Growth, Renewable Resources and Pollution: Thresholds and Cycles.” Journal of Economic Dynamics & Control 28: 1149–1157.10.1016/S0165-1889(03)00077-0Search in Google Scholar

Published Online: 2019-04-09

©2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 22.2.2024 from
Scroll to top button