Pollution, carrying capacity and the Allee effect

Stefano Bosi; David Desmarchelier

doi:10.1515/snde-2019-0016

Published by De Gruyter April 9, 2019

Pollution, carrying capacity and the Allee effect

Stefano Bosi and David Desmarchelier

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2019-0016

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Abstract

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.

Keywords: Allee effect; carrying capacity; Hopf bifurcation; logistic dynamics; pollution; Ramsey model

JEL Classification: E32; O44

Appendix

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=e−θtu(c,N)+λ[(r−δ)h+w−c], deriving the first-order conditions ∂H/∂c=0, ∂H/∂h=−λ˙ and ∂H/∂λ=h˙, and defining μ≡λeθt, 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

(28)0<εcεN<−τNi′(τ)Ni(τ)τc′(τ)c(τ)

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

J≡[∂f1∂μ∂f1∂k∂f1∂N∂f1∂P∂f2∂μ∂f2∂k∂f2∂N∂f2∂P∂f3∂μ∂f3∂k∂f3∂N∂f3∂P∂f4∂μ∂f4∂k∂f4∂N∂f4∂P]=[0(θ+δ)(1−α)μk00−cμθ−cN000φψNP0(b−mτ)θ+δ1−τ0−a]

where

φ≡[NA(P)−1][1−NC(P)]+NA(P)[1−NC(P)]−NC(P)[NA(P)−1]

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

[NA(P)−1][1−NC(P)]=0

We observe that

φ1≡φ(N1)=1−A(P)C(P)φ2≡φ(N2)=1−C(P)A(P)ψ1≡ψ(N1)=εA[A(P)C(P)−1]ψ2≡ψ(N2)=εC[C(P)A(P)−1]

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

cμ=−1εcμ=−ck1εkμ=−πεkμcN=η(1−ε)εcN=ckη(1−ε)εkN=πη(1−ε)εkN

Thus, we obtain

J≡[∂f1∂μ∂f1∂k∂f1∂N∂f1∂P∂f2∂μ∂f2∂k∂f2∂N∂f2∂P∂f3∂μ∂f3∂k∂f3∂N∂f3∂P∂f4∂μ∂f4∂k∂f4∂N∂f4∂P]=[0(θ+δ)(1−α)μk00γεkμθ−γη(1−ε)εkN000φψNP0(b−mτ)θ+δ1−τ0−a]

□

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

S3T=S2±S22−4D2

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

S3(η−)T=12[S2−S22−4D]S3(η+)T=12[S2+S22−4D]

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. □

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Published Online: 2019-04-09

Pollution, carrying capacity and the Allee effect

Abstract

Appendix

References

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