Biodiversity, Infectious Diseases, and the Dilution Effect

Abstract

Biologists point out that biodiversity loss contributes to promote the transmission of diseases. In epidemiology, this phenomenon is known as dilution effect. Our paper aims to introduce this effect in an economic model where the spread of an infectious disease is considered. More precisely, we embed a SIS model into a Ramsey model (1928) where a pollution externality coming from production affects the evolution of biodiversity. Biodiversity is assimilated to a renewable resource and affects the infectivity of the disease (dilution effect). A green tax is levied on production at the firm level to finance depollution according to a balanced budget rule. In the long run, a disease-free and an endemic regime are possible. We focus only on the second case and we find that the magnitude of the dilution effect determines the number of steady states. When the dilution effect remains low, there are two cases depending on the environmental impact of production: (1) a low impact implies two steady states with high and low biodiversity respectively; (2) a large impact rules out any steady state. Conversely, when the dilution effect becomes high, a (unique) steady state always exists: a strong dilution effect works as a buffer and prevents the human pressure from being lethal for biodiversity in the long run. Moreover, under a low dilution effect, a higher green tax rate always impairs biodiversity at the low steady state, while this green paradox is over under a high dilution effect. In the short run, we show that a limit cycle can arise around the high biodiversity steady state when the dilution effect is low. Surprisingly, the limit cycle is preserved under a high dilution effect. In other words, even if a strong dilution effect preserves the biodiversity in the long run and prevents the economy from the green paradox, it does not shelter the economy from the occurrence of biodiversity fluctuations.

1 Introduction

During the past decades, the number of infectious diseases outbreaks has increased worldwide as well as the number of emerging infectious diseases [31]. Diseases are human-specific when contagions are only between humans and they are zoonotic when contagions are from animals to humans. Following Lloyd-Smith et al. [23], zoonoses represent 60% to 76% of emerging diseases and, in this respect, they constitute a major concern for public health. Moreover, since 1970, the World Wide Fund for Nature [32] reports that the size of population of vertebrate species has been divided by two. A negative relation seems to link the emergence of infectious diseases (in particular zoonoses) and biodiversity. In the literature, this negative link is known as dilution effect.¹

The dilution effect can be simply explained. According to Civitello et al. [11], a high level of biodiversity inhibits the proliferation of parasites and prevents the spread of infectious diseases. An interesting example of dilution effect is reported by Keesing et al. [20] about the Lyme disease transmission in the northeastern USA. Humans contract the Lyme disease by the bite of an infected blacklegged tick (Ixodes scapularis). The blacklegged ticks become infected when they feed on infected hosts which are primarily the white-footed mouse (Peromyscus leucopus) and the Virginia opossum (Didelphis virginiana). Interestingly, blacklegged ticks feeding on the Virginia opossum are much more likely to be uninfected than the ones feeding on the white-footed mouse. This is due to the fact that the Virginia opossum is a poor host for the Lyme disease while the white-footed mouse is a much better host. As reported in Keesing et al. [20], the Virginia opossum acts as a buffer for the Lyme disease and deflects away the disease from the white-footed mouse. That is, the reported reduction in the population of Virginia opossums promotes the Lyme diseases. In this example, a lower biodiversity level promotes the Lyme disease: a dilution effect occurs.

According to Keesing et al. [20], the dilution effect was observed for various infectious diseases such as hantavirus disease, malaria, schistosomiasis, or West Nile fever. However, as reported by Civitello et al. [11], the generality of the dilution effect remains controversial in the literature. On the one hand, as seen above, biodiversity loss increases the pathogen’s concentration in the remaining species which can increase the disease transmission (dilution effect), especially when the remaining species are good hosts. However, on the other hand, as reported by Salkeld et al. [29], ecosystems with high biodiversity can be richer in parasite diversity which can promote disease transmission. Keesing et al. [19] show that, if the net overall effect of biodiversity on disease risk is negative, a dilution effect occurs while, if the net overall effect is positive, an amplification effect takes place. Therefore, the debate is about the sign of the overall net effect (positive or negative) and the generality of the dilution effect.

In their study, Keesing et al. [20] advocate for the generality of the dilution effect. This conclusion is also supported by recent empirical evidences by Civitello et al. [11]. They show broad evidences for the dilution effect and conclude that anthropogenic declines in biodiversity increase human and wildlife diseases. The existence and the generality of the dilution effect is of great importance because it means that environmental and biodiversity preservation allows to preserve human health. This complementarity reveals also interdependences between biodiversity and the whole economy. Indeed, illness is recognized as an important source of both productivity loss ([25]; or [14]) and workers absenteeism ([1]; and [14]). That is, economic activities pollute and imply both climate change and biodiversity loss which promotes in turn infectious diseases (dilution effect) and impairs economic activities (productivity loss, workers absenteeism). In this paper, we aim to develop a theoretical framework to take into account this complementarity between biodiversity and economic activities induced by the dilution effect. To this purpose, an interdisciplinary approach is needed.

In epidemiology, there is a panoply of theoretical models to study the spread of an infectious disease through a population [17]. To keep things as simple as possible, we have decided to consider the simplest model to represent the change in the share of healthy people over time: the susceptible–infective–susceptible (SIS) model. In this stylized epidemiological framework, two parameters capture the dynamics: (1) the probability of a susceptible individual to become ill after a contact with an infected individual and (2) the recovery rate driving the lapse of time the infected individual spends to recover from the disease. Dynamics are straightforward. A disease-free steady state coexists with an endemic one. When (1) exceeds (2), the endemic steady state is stable while the other, unstable.

To bridge the gap between economy, ecology, and epidemiology, we propose to embed the SIS model into a Ramsey growth model where the production activities pollute and impair the biodiversity. The Ramsey framework is better than an overlapping generations (OLG) model to represent short-run epidemiological dynamics. Indeed, in a two-period OLG model, a period covers the half-life of an individual, says 35 years according to the average expectancy of life across the world. Such a long period is inappropriate to reproduce the short-run cycles of most common diseases.

To simplify the architecture, we assimilate biodiversity to a renewable resource affecting the household’s immune system. We introduce a two-sided dilution effect assuming the probability to become ill as a decreasing function of biodiversity and the recovery rate as an increasing function. As in Goenka et al. [16], we consider a labor force constituted only of healthy people. We introduce a government who levies a simple Pigouvian tax on production at the firm level in order to finance depollution according to a balanced budget rule. Such a unified framework gives us the opportunity to observe the effects of environmental policies on economic variables, the biodiversity, and the pollution level.

The integration of the SIS model into a Ramsey model is not new and dates back to Goenka and Liu [15]. They have considered a discrete-time model in which healthy people tune their labor supply through a consumption–leisure arbitrage. In a continuous version of the model where labor supply is exclusively driven by the number of healthy people, Goenka et al. [16] address the issue of optimal health expenditures. More recently, Bosi and Desmarchelier [9] have reconsidered the continuous-time version developed by Goenka et al. [16] to take in account the interplay between a flow of pollution and infectious diseases. Bosi and Desmarchelier [9] have pointed out that, when pollution becomes excessive, two limit cycles can appear (stable and unstable) near the endemic steady state through a Hopf bifurcation.

We study the equilibrium either in the long or the short run. In the long run, as in the standard SIS model, a disease-free regime coexists with the endemic one. The disease-free regime is characterized by the existence of two steady states: one experiences a low biodiversity level; the other, a high level. Interestingly, when the environmental impact of production becomes excessive, the two steady states collide and disappear. This case captures the possibility of a mass extinction under human pressure on the environment. In the endemic regime, under a low dilution effect, we recover the key features of the disease-free regime. Conversely, under a strong dilution effect, there is always a unique steady state whatever the environmental impact of production. Therefore, very importantly, the complementarity between biodiversity and production activities induced by a large dilution effect seems to prevent the occurrence of a mass extinction. Moreover, a paradox emerges under a moderate dilution effect at the steady state with low biodiversity: a higher green tax rate lowers the biodiversity. This counter-intuitive effect is similar to the static green paradox pointed out by Bosi and Desmarchelier [6].² Conversely, a green tax rate always increases the biodiversity level at the steady state with higher biodiversity. Interestingly, the static green paradox is ruled out by a strong dilution effect. In the short run, we show that both the dilution effects (low and high) are compatible with the existence of a limit cycle (arising through a Hopf bifurcation) around the high biodiversity steady state when preferences exhibit a complementarity between biodiversity and consumption. To sum up, a high dilution effect seems to have a double benefit: (1) it preserves biodiversity in the long run and (2) it prevents the economy from the green paradox. Nevertheless, it is not able to avoid the fluctuations of biodiversity (limit cycle) around the steady state.

The rest of the paper is organized as follows. Section 2 introduces the model. Sections 3 derives the equilibrium system. Sections 4 and 5 focus on the long- and short-run dynamics. A numerical illustration with isoelastic fundamentals is provided in Section 6. Section 7 concludes.

2 Model

In this paper, we are interested in the relationship between biodiversity loss and transmission of infectious diseases in an economic context. Our mathematical approach, despite some technical aspects, leads to a deep understanding of economic and ecological feedbacks, and unveils their practical consequences. In the following section, we introduce the general framework to develop in the rest of the paper.

2.1 Disease

Epidemiologists use the SIS model to study the spread of endemic diseases. Population (N_t) is divided in two classes: susceptible (S_t) and infective (I_t) with S_t + I_t = N_t. We consider a wide range of infectious diseases, not a specific one. So the infective class covers many different illnesses. The proportion of susceptible and infective are given by s_t = S_t/N_t and i_t = I_t/N_t. β > 0 denotes the average number of adequate contacts (sufficient to transmit the disease) of an infective per unit of time, and S_t/N_t the probability to face a susceptible during a contact. β increases in the transmissibility due to the virulence and pathogenicity of microbes, which increases in turn in the loss of biodiversity. Thus, βS_t/N_t is the average number of adequate contacts with susceptibles of one infective per unit of time, while the number of new infectives per unit of time is given by βI_tS_t/N_t. An infective is seek during a period of time after which he recovers and becomes a new susceptible ($\gamma =-\dot {I}_{t}/I_{t}$ is the recovery rate in absence of new contamination, a sort of exponential decay rate from infection). The recovery rate decreases with the virulence and, so, with the loss of biodiversity. Notice that the SISmodel postulates that the infection does not confer immunity.

In an oversimplified world with no births, no deaths, and no migrations, the population remains constant over time ($\dot {N}_{t}=0$). The evolution of S_t and I_t over time is simply given by the following:

$$ \begin{array}{@{}rcl@{}} \dot{S}_{t} &=& -\beta\frac{I_{t}}{N_{t}}S_{t}+\gamma I_{t}, \end{array} $$

(1)

$$ \begin{array}{@{}rcl@{}} \dot{I}_{t} &=& \beta\frac{I_{t}}{N_{t}}S_{t}-\gamma I_{t}\text{.} \end{array} $$

(2)

Since $\dot {N}_{t}=0$, it follows that $\dot {S}_{t}+\dot {I}_{t}=0$ and Eq. 1 becomes the following:

$$ \dot{s}_{t}=\left( 1-s_{t}\right) \left( \gamma-\beta s_{t}\right) \text{.} $$

(3)

As in Goenka et al. [16], we assume that the labor force (L_t) consists only of healthy people:L_t = S_t. Since l_t = L_t/N_t ≤ 1, l inherits the dynamics of s:

$$ \dot{l}_{t}=\left( 1-l_{t}\right) \left( \gamma-\beta l_{t}\right) \text{.} $$

(4)

We can see that Eq. 4 exhibits two steady states: l = 1 and l = l^∗ = γ/β. The first one is called disease-free because the disease disappears while the other is called endemic because the disease persists. Clearly, the endemic steady state becomes meaningless when γ > β because in this case l^∗ > 1. In other words, the endemic steady state is admissible from an economic point of view if and only if γ < β.

As empirically observed by Keesing et al. [20] among others, a biodiversity loss promotes infectious diseases. This is the dilution effect. The goal of our model is to understand its macroeconomic consequences. Since the dilution effect captures a negative correlation between the disease prevalence and the biodiversity level, we assume that a higher biodiversity level (B_t) reduces the disease transmissibility (β) and increases the recovery rate (γ). The following hypotheses sum up this idea.

Assumption 1

$\beta ^{\prime }\left (B_{t}\right ) <0$ and $\gamma ^{\prime }\left (B_{t}\right ) >0$ with $\lim _{B_{t} \rightarrow 0}\beta \left (B_{t}\right ) =\infty$, $\lim _{B_{t}\rightarrow \infty }\beta \left (B_{t}\right ) =0$, $\lim _{B_{t}\rightarrow 0}\gamma \left (B_{t}\right ) =0$ and $\lim _{B_{t}\rightarrow \infty }\gamma \left (B_{t}\right ) =\infty$.

We introduce a new formal definition of dilution effect, encompassing both the impacts of biodiversity on the disease transmissibility (β) and on the recovery rate (γ).

Definition 1 (dilution effect)

The dilution effect is given by the following:

$$ d\left( B_{t}\right) \equiv\varepsilon_{\gamma}\left( B_{t}\right) -\varepsilon_{\beta}\left( B_{t}\right) >0, $$

(5)

where

$$ \varepsilon_{\beta}\left( B_{t}\right) \equiv\frac{B_{t}\beta^{\prime }\left( B_{t}\right) }{\beta\left( B_{t}\right) }\!<\!0\text{ and } \varepsilon_{\gamma}\left( B_{t}\right) \!\equiv\!\frac{B_{t}\gamma^{\prime }\left( B_{t}\right) }{\gamma\left( B_{t}\right) }>0, $$

(6)

are the first-order elasticities.

We observe that, at the endemic steady state, the dilution effect captures also the sensitivity (elasticity) of labor supply with respect to biodiversity, that is, HCode $d\left (B_{t}\right ) =B_{t}l^{\prime }\left (B_{t}\right ) /l\left (B_{t}\right ) >0$. As we will see later, the dilution effect has serious consequences on epidemiological and economic dynamics. To simplify the presentation, we normalize the population size to the unity: N_t = 1.

2.2 Preferences

The household earns a capital income r_th_t and a labor income $\bar {\omega }_{t}$, where r_t and h_t denote respectively the real interest rate and the individual wealth at time t. Income is consumed(c_t), saved, and invested according to the budget constraint:

$$ \dot{h}_{t}\leq\left( r_{t}-\delta\right) h_{t}+\bar{\omega}_{t} -c_{t}\text{.} $$

(7)

In this model, healthy people work while sick people do not. However, for simplicity, we assume a perfect social security, that is, a full unemployment insurance in the case of illness. Healthy and sick agents earn the same labor income $\bar {\omega }_{t}$. L_t healthy people supply one unit of labor at a wage w_t. Under a balanced budget rule for social security, we obtain $\bar {\omega }_{t}N_{t}=w_{t}L_{t}$. Therefore, $\bar {\omega }_{t}=w_{t}l_{t}$.

Gross investments include the capital depreciation at the rate δ. Since N_t = 1, we obtain L_t = N_tl_t = l_t, K_t = N_th_t = h_t, and h_t = K_t/N_t = k_tl_t.

Let $u\left (c_{t},B_{t}\right )$ be the utility function of the representative household. We assume that biodiversity affects marginal utility of consumption (u_cB ≠ 0) which means that it affects the consumption behavior. Indeed, biodiversity has not only a positive impact on physical health through the dilution effect but has also a positive influence on mental health. For instance, as reported by Dean et al. [12], biodiversity in cities has some psychosocial benefits: recovery from stress, self-regulation of emotions, restoration of attention fatigue, and enhanced sense of community. In their study, Dean et al. [12] point out that these psychosocial benefits preserve mental health and prevent a depressive behavior. Even if, to the best of our knowledge, there is no empirical evidence about the effect of biodiversity on consumption demand, its benefits on mental health suggest also a role in consumption. Indeed, less biodiversity makes agents more depressive and can reduce their consumption demand (u_cB > 0). Conversely, they can compensate the loss of pleasure by increasing their consumption demand (u_cB < 0). The ambiguous effects of biodiversity on this demand remind us those of pollution on consumption highlighted by [24]. The following assumption sums up the properties of the utility function.

Assumption 2

Preferences are rationalized by a non-separable utility function $u\left (c_{t},B_{t}\right )$. First- and second-order restrictions hold on the sign of derivatives: u_c > 0, u_B > 0, and u_cc < 0, and the limit conditions:

$$ \lim_{c_{t}\rightarrow0^{+}}u_{c}\left( c_{t},B_{t}\right) =\infty\text{ and }\lim_{c_{t}\rightarrow+\infty}u_{c}\left( c_{t} ,B_{t}\right) =0\text{.} $$

We introduce the second-order elasticities:

$$ \left[ \begin{array} [c]{cc} \varepsilon_{\mathit{cc}} & \varepsilon_{\mathit{cB}}\\ \varepsilon_{\mathit{Bc}} & \varepsilon_{\mathit{BB}} \end{array} \right] \equiv\left[ \begin{array} [c]{cc} \frac{c_{t}u_{\mathit{cc}}}{u_{c}} & \frac{B_{t}u_{\mathit{cB}}}{u_{c}}\\ \frac{c_{t}u_{\mathit{Bc}}}{u_{B}} & \frac{B_{t}u_{\mathit{BB}}}{u_{B}} \end{array} \right], $$

(8)

− 1/ε_cc represents the intertemporal elasticity of substitution in consumption while ε_cB captures the effect of biodiversity on the marginal utility of consumption. Typically, if ε_cB > 0 (< 0), biodiversity and consumption are complements (substitutes) for households.

The illness lowers labor supply and the individual income in turn. The agent maximizes the intertemporal utility function as follows:

$$ {\int}_{0}^{\infty}e^{-\theta t}u\left( c_{t},B_{t}\right) \text{dt}, $$

(9)

under the budget constraint (7), where 𝜃 > 0 is the rate of time preference.

Proposition 2

The first-order conditions of the consumer’s program are given by a static relation as follows:

$$ \mu_{t}=u_{c}\left( c_{t},B_{t}\right), $$

(10)

a dynamic Euler equation and the budget constraint (1), now binding:

$$ \begin{array}{@{}rcl@{}} \dot{\mu}_{t} & =&\mu_{t}\left( \theta+\delta-r_{t}\right), \end{array} $$

(11)

$$ \begin{array}{@{}rcl@{}} \dot{h}_{t} & =&\left( r_{t}-\delta\right) h_{t}+w_{t}l_{t}-c_{t}, \end{array} $$

(12)

jointly with the transversality condition $\lim _{t\rightarrow \infty }e^{-\theta t}\mu _{t}h_{t}=0$. μ_t denotes the multiplier associated to the budget constraint.

Proof

See the Appendix. □

Applying the implicit function theorem to the static relation $\mu _{t} =u_{c}\left (c_{t},B_{t}\right )$, we obtain the consumption function $c_{t}\equiv c\left (\mu _{t},B_{t}\right )$ with elasticities as follows:

$$ \frac{\mu_{t}}{c_{t}}\frac{\partial c}{\partial\mu_{t}}=\frac{1} {\varepsilon_{\mathit{cc}}}<0\text{ and }\frac{B_{t}}{c_{t}}\frac{\partial c}{\partial B_{t}}=-\frac{\varepsilon_{\mathit{cB}}}{\varepsilon_{\mathit{cc}}}, $$

(13)

where ε_cB captures the effect of biodiversity on consumption demand. More precisely, when the household’s preferences display complementarity (substituability) between biodiversity and consumption, a higher biodiversity level entails a higher (a lower) consumption demand.

To sum up, biodiversity affects the households in two respects: (1) through the labor income (a lower biodiversity implies more persistent infectious diseases (dilution effect) lowering the labor supply and the labor income in turn) and (2) directly through the utility function ($\varepsilon _{\text {cB}}\lessgtr 0$).

2.3 Production

A single firm³ chooses the amount of capital (K_t) and labor (L_t) to maximize the profit taking as given the real interest rate r_t as well as the wage rate w_t. In addition, the government levies a proportional tax $\tau \in \left (0,1\right )$ on polluting production $F\left (K_{t},L_{t}\right )$ to finance depollution expenditures. The following assumption sums up the properties of the production function.

Assumption 3

The production function $F:\mathbb {R}_{+}^{2}\rightarrow \mathbb {R}_{+}$ is C², homogeneous of degree one, strictly increasing and concave. Inada conditions hold.

The profit maximization $\max_{K_{t},L_{t}}\left [ F\left (K_{t} ,L_{t}\right ) -r_{t}K_{t}-w_{t}L_{t}-\tau F\left (K_{t},L_{t}\right ) \right ]$ entails the following first-order conditions:

$$ r_{t}=\left( 1-\tau\right) \rho\left( k_{t}\right) \text{ and } w_{t}=\left( 1-\tau\right) \omega\left( k_{t}\right), $$

(14)

where k_t ≡ K_t/L_t is the capital intensity and $f\left (k_{t}\right ) \equiv F\left (k_{t},1\right )$ the average productivity, and $\rho \left (k_{t}\right ) \equiv f^{\prime }\left (k_{t}\right )$ and $\omega \left (k_{t}\right ) \equiv f\left (k_{t}\right ) -k_{t}f^{\prime }\left (k_{t}\right )$.

We introduce the capital share in total disposable income $\alpha \left (k_{t}\right ) \equiv k_{t}f^{\prime }\left (k_{t}\right ) /f\left (k_{t}\right )$ and the elasticity of capital-labor substitution $\sigma \left (k_{t}\right ) \equiv \alpha \left (k_{t}\right ) \omega \left (k_{t}\right ) /\left [ k_{t}\omega ^{\prime }\left (k_{t}\right ) \right ]$. We obtain the elasticities of factor prices: $k_{t}\rho ^{\prime }\left (k_{t}\right ) /\rho \left (k_{t}\right ) =\left [ \alpha \left (k_{t}\right ) -1\right ] /\sigma \left (k_{t}\right )$ and $k_{t}\omega ^{\prime }\left (k_{t}\right ) /\omega \left (k_{t}\right ) =\alpha \left (k_{t}\right ) /\sigma \left (k_{t}\right )$.

2.4 Government

The government uses all the tax revenues to finance depollution expenditures (G_t) according to a balanced budget rule:

$$ G_{t}=\tau F\left( K_{t},L_{t}\right) \text{.} $$

(15)

Despite the simplicity of this tax scheme, we will see that tax rate τ has important environmental effects in the long run.

2.5 Biodiversity

To keep things as simple as possible, we assimilate the biodiversity to a renewable natural resource. Following Ayong Le Kama [3] and Wirl [33], the dynamics of natural resource is given by the following:

$$ \dot{B}_{t}=g\left( B_{t}\right) -P_{t}, $$

(16)

where $g\left (B_{t}\right )$ and P_t represent the reproduction function and the pollution level respectively. In the following, we will refer to Eq. 16 as reproduction function in a broad sense. In the spirit of Wirl [33] and Bella [5], we specify $g\left (B_{t}\right )$ as a Pearl-Verhulst logistic function: $g\left (B_{t}\right ) \equiv B_{t}\left (1-B_{t}\right )$. Clearly, if B_t < 1, then $g\left (B_{t}\right ) >0$ and $\dot {B}_{t}\lessgtr 0$ depending on the pollution level (P_t) while B_t > 1 implies $g\left (B_{t}\right ) <0$ and then $\dot {B}_{t}<0$.

Interestingly, since $g^{\prime }\left (B_{t}\right ) =1-2B_{t}$, the maximal sustainable yield occurs at B_t = 1/2. In other words, for every B_t > 1/2, g decreases, which means that B_t is bounded and cannot tend to infinity. Interestingly, Wirl [33] has pointed out that limit cycles can occur if and only if B_t < 1/2 (the maximal sustainable yield) at the steady state.

To simplify the presentation, we assume as in Itaya [18] or in Fernandez et al. [13] that pollution is a flow coming from production activity as follows:

$$ P_{t}=\mathit{aY}_{t}-\mathit{bG}_{t}, $$

(17)

where a and b capture respectively the environmental impact of production and the depollution efficacy.

Considering Eqs. 15, 16, and 17, we find the natural resource accumulation law as follows:

$$ \dot{B}_{t}=B_{t}\left( 1-B_{t}\right) +\left( b\tau-a\right) F\left( K_{t},L_{t}\right) \text{.} $$

(18)

A non-negative net pollution requires an additional assumption.

Assumption 4

a > bτ.

3 Equilibrium

Since N_t = 1, the natural resource accumulation law becomes $\dot {B}_{t}=B_{t}\left (1-B_{t}\right ) +\left (b\tau -a\right ) l_{t}f\left (k_{t}\right )$. At the equilibrium, all the markets clear. This leads to the following proposition.

Proposition 3

Dynamics are driven by a four-dimensional dynamic system:

$$ \begin{array}{@{}rcl@{}} \dot{\mu}_{t} & =&\mu_{t}\left[ \theta+\delta-\left( 1-\tau\right) \rho\left( k_{t}\right) \right], \end{array} $$

(19)

$$ \begin{array}{@{}rcl@{}} \dot{k}_{t} & =&\left[ \left( 1-\tau\right) \rho\left( k_{t}\right) -\delta\right] k_{t}+\left( 1-\tau\right) \omega\left( k_{t}\right)\\ &&-\frac{c\left( \mu_{t},B_{t}\right) }{l_{t}}-k_{t}z\left( l_{t}, B_{t}\right), \end{array} $$

(20)

$$ \begin{array}{@{}rcl@{}} \dot{l}_{t} & =&l_{t}z\left( l_{t},B_{t}\right), \end{array} $$

(21)

$$ \begin{array}{@{}rcl@{}} \dot{B}_{t} & =&B_{t}\left( 1-B_{t}\right) +\left( b\tau-a\right) l_{t}f\left( k_{t}\right), \end{array} $$

(22)

with $z\left (l_{t},B_{t}\right ) \equiv \left [ \gamma \left (B_{t}\right ) -\beta \left (B_{t}\right ) l_{t}\right ] \left (1-l_{t}\right ) /l_{t}$, jointly with the transversality condition.

Proof

See the Appendix. □

Equations 19–22 is a bioeconomic system. More precisely, Eqs. 19 and 20 capture the economic part (the Ramsey model) while Eqs. 21 and 22 represent respectively the epidemiological and the ecological part of the model. Figure 1 summarizes all relations between firms, households, biodiversity, and the government.

Fig. 1

Model structure

In the next sections, we will analyze the behavior of the dynamical system (19)–(22) to observe how the dilution effect affects the economy in the long run (steady state) and in the short run (local dynamics).

4 Steady State

At the steady state, all the variables remain constant: $\dot {\mu }_{t}=\dot {k}_{t}=\dot {l}_{t}=\dot {B}_{t}=0$. From Eq. 19, we obtain the modified golden rule (MGR):

$$ \rho\left( k^{\ast}\right) =\frac{\theta+\delta}{1-\tau}\text{.} $$

(23)

As usual, the MGR gives the capital intensity k^∗ at the steady state. Indeed, Assumption 3 ensures the invertibility of ρ. Thus, the capital intensity at the steady state is given by $k^{\ast }=\rho ^{-1}\left (\left (\theta +\delta \right ) /\left (1-\tau \right ) \right ) >0$. It is interesting to remark that both biodiversity and infectious disease have no effects on the capital intensity in the long run. Focus now on Eq. 21. At the steady state, $z\left (l,B\right ) =0$, that is, $\left (1-l\right ) \left [ \gamma \left (B\right ) -\beta \left (B\right ) l\right ] =0$ with solutions, l = 1 or $l\equiv l^{\ast }=\gamma \left (B\right ) /\beta \left (B\right )$. Hence, we recover one of the main feature of the SIS model; indeed, two steady states coexist: l = 1 is the disease-free steady state, while l = l^∗ is the steady state with an endemic disease.

Since $z\left (l,B\right ) =0$ at the steady state, Eq. 20 gives simply the consumption level as follows:

$$ c^{\ast}=l\left[ \left( 1-\tau\right) f\left( k^{\ast}\right) -\delta k^{\ast}\right], $$

(24)

with l = 1 or l = l^∗.

According to Eq. 24, we see that biodiversity and the infectious disease affect the consumption level in the long run when l = l^∗, even if (23) shows that the capital intensity of steady state is not affected: because of the dilution effect, a decrease in biodiversity implies a reduction in labor supply entailing in turn a drop in consumption at the steady state.

We know that k^∗ is unique and positive at the steady state. Thus, given l, according to Eq. 24, there is a unique and positive value c^∗ of c at the steady state. Moreover, Eq. 10 implies that, given c^∗ and B, there is a unique and positive shadow price μ^∗.

Finally, at the steady state, the natural resource accumulation law (22) becomes the following:

$$ B^{2}-B+\left( a-b\tau\right) lf\left( k^{\ast}\right) =0, $$

(25)

with l = 1 or l = l^∗.

The existence and the uniqueness/multiplicity of the endemic steady state l^∗ (as well as of the disease-free steady state) depend upon the number of B satisfying Eq. 25. In the following, we compare the disease-free and the endemic regime.

4.1 Disease-Free Steady State

At the disease-free steady state, the disease no longer exists and all the labor force is employed, that is, l = 1. In this case, Eq. 25 becomes the following:

$$ g\left( B\right) \equiv B\left( 1-B\right) =\left( a-b\tau\right) f\left( k^{\ast}\right) \text{.} $$

(26)

At the disease-free steady state, B satisfies Eq. 26. Under Assumption 4, $B\in \left (0,1\right )$ at the steady state. Moreover, given k^∗, the environmental impact of production (a) drives the number of steady state. Indeed consider (26), when $a\rightarrow +\infty$, the RHS never crosses the LHS while, when $a\rightarrow 0^{+}$, the RHS crosses the LHS for two distinct values of B. Of course, multiple steady states arise because of the bell-shaped reproduction function $g\left (B\right )$.

Let us set a^∗ the threshold value of a:

$$ a^{\ast}\equiv b\tau+\frac{1}{4f\left( k^{\ast}\right) }, $$

The roots of Eq. 26 are simply given by the following::

$$ \begin{array}{@{}rcl@{}} B_{1} & \equiv&\frac{1}{2}\left( 1-\sqrt{\Delta}\right), \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} B_{2} & \equiv&\frac{1}{2}\left( 1+\sqrt{\Delta}\right), \end{array} $$

(28)

where ${\Delta }\equiv 1-4\left (a-b\tau \right ) f\left (k^{\ast }\right )$. Distinct biodiversity levels B₁ and B₂ exist if and only if a < a^∗ (or, equivalently, Δ > 0). In this case, 0 < B₁ < 1/2 < B₂ < 1. The following proposition sums up these results.

Proposition 4

Let l = 1.

• (1) If a < a^∗, there are two steady states with 0 < B₁ < 1/2 < B₂ < 1.

• (2) If a > a^∗, there are no steady states.

And, if a = a^∗, B₁ = B₂ = 1/2.

a represents the environmental impact of production and reflects the human pressure on nature. When the pressure exceeds the threshold (a > a^∗), biodiversity fails to exist in the long run (there are no steady states). The dynamic properties of this case, where the disease does not persist in the long run (l = 1), has been studied by Bosi and Desmarchelier [10].

4.2 Endemic Steady State

Focus on the endemic steady state⁴ (that is, $l=l^{\ast }\equiv \gamma \left (B\right ) /\beta \left (B\right )$). From Eq. 25, the stationary biodiversity level satisfies the following:

$$ \varphi\left( B\right) \equiv B\left( 1-B\right) \frac{\beta\left( B\right) }{\gamma\left( B\right) }=\left( a-b\tau\right) f\left( k^{\ast}\right) >0, $$

(29)

which implies $B\in \left (0,1\right )$.

For simplicity, in order to obtain a dilution effect independent of the endogenous biodiversity level B, focus on the following isoelastic functions:

$$ \beta\left( B_{t}\right) \equiv A_{\beta}B_{t}^{\varepsilon_{\beta}}\text{ and }\gamma\left( B_{t}\right) \equiv A_{\gamma}B_{t}^{\varepsilon_{\gamma} }, $$

(30)

with A_β,A_γ > 0, ε_β < 0, and ε_γ > 0 (see Eq. 6). According to Eqs. 5 and 29, we get d = ε_γ − ε_β and

$$ \varphi\left( B\right) =\left( 1-B\right) B^{1-d}\frac{A_{\beta} }{A_{\gamma}}\text{.} $$

(31)

The first-order elasticity of φ is given by the following:

$$ \frac{B\varphi^{\prime}\left( B\right) }{\varphi\left( B\right) } =\frac{1-2B}{1-B}-d\text{.} $$

(32)

Let

$$ \bar{B}\equiv\frac{1-d}{2-d}\text{ and }\bar{a}\equiv b\tau+\frac {\varphi\left( \bar{B}\right) }{f\left( k^{\ast}\right) }, $$

(33)

where k^∗ is the capital of MGR. When the dilution effect is low (d < 1), $\varphi ^{\prime }\left (B\right ) >0$ if and only if $B<\bar {B}$. Thus, when the dilution effect is low, φ is bell-shaped with $\varphi \left (0\right ) =\varphi \left (1\right ) =0$ and a maximum at B = $\bar {B}\in \left (0,1\right )$. According to Eq. 29, two steady states, say B₃ and B₄ with B₃ < B₄, exist if and only if

$$ \left( a-b\tau\right) f\left( k^{\ast}\right) <\varphi\left( \bar {B}\right), $$

(34)

or, equivalently, if and only if $a<\bar {a}$. Clearly, $0<B_{3}<\bar {B} <B_{4}<1$. In other words, when the dilution effect is low, two steady state exist, if and only if the environmental impact of production is not too high ($a<\bar {a}$). Conversely, the steady state fails to exist when the environmental impact of production is excessive ($a>\bar {a}$). Of course, the steady state becomes unique ($B_{3}=B_{4}=\bar {B}$) when $a=\bar {a}$. Let us provide a formal statement.

Proposition 5 (small dilution effect)

Focus on the isoelastic case (30) with 0 < d < 1 and the endemic steady state l = l^∗.

• (1) If $a<\bar {a}$, there are two steady states B₃ and B₄ such that $0<B_{3}<\bar {B}<B_{4}<1$.

• (2) If $a=\bar {a}$, there is a unique steady state $\bar {B}$.

• (3) If $a>\bar {a}$, there are no steady states.

Thus, under a low dilution effect (d < 1), we recover in the endemic regime (l = l^∗) the properties of the disease-free regime (l = 1) (Proposition 4): when human pressure on nature becomes excessive (a > a^∗), the steady state vanishes and the biodiversity disappears in the long run.

The parallel between these two regimes existing under a low dilution no longer holds when the dilution effect becomes high (d > 1). Indeed, according to Eqs. 32 and 33, we find $\varphi ^{\prime }\left (B\right ) <0$ for any $B\in \left (0,1\right )$ with $\lim _{B\rightarrow 0}\varphi \left (B\right ) =+\infty$ and $\lim _{B\rightarrow 1}\varphi \left (B\right ) =0$. Hence, there is a unique solution of Eq. 29, say B₅.

Proposition 6 (large dilution effect)

Consider the endemic steady state l = l^∗ in the isoelastic case (30) with d > 1. Then, there exists a unique steady state $B_{5}\in \left (0,1\right )$.

How could we explain the different consequences of low and high dilution effects? As seen in Proposition (5), the steady state fails to exist under a low dilution effect (d < 1) if the environmental impact of production becomes excessive ($a>\bar {a}$). Production activities are so bad for environment that biodiversity no longer survives in the long run. Conversely, under a strong dilution effect (d > 1), a slight decrease of biodiversity has a large negative effect on labor supply and production in turn. Thus, a strong dilution effect works as a buffer and prevents the human pressure from being lethal for biodiversity in the long run. The dilution effect introduces a kind of complementarity between the level of economic activity and nature: when d > 1, the complementarity becomes sufficiently powerful to preserve the biodiversity from a mass extinction. Since production affects biodiversity through a flow of pollution, we can reasonably expect that the effectiveness of a green tax rate (τ) largely depends on the magnitude of dilution.

4.3 Comparative Statics

In this section, we are interested in long-run impacts of the green tax rate on both economic and biological variables at the endemic steady state (l = l^∗). The next proposition sums up the results under low (d < 1) and high (d > 1) dilution effect.

Proposition 7

Let Assumption 4 hold and $B\in \left (0,1\right )$. Focus on the qualitative impact of τ on the endemic steady state (l = γ/β). We have the following:

$$ \frac{\tau}{k^{\ast}}\frac{dk^{\ast}}{d\tau}<0\text{.} $$

(35)

Moreover,

• (1) if 0 < d < 1,

  $$ \frac{\tau}{B_{3}}\frac{\text{dB}_{3}}{d\tau} \!<\!0\text{ and }\frac{\tau}{B_{4}}\frac{dB_{4}}{d\tau}>0,$$
  $$ \frac{\tau}{l^{\ast}}\frac{\text{dl}^{\ast}}{d\tau} \! <\!0\text{ if }B=B_{3}\text{, and }\frac{\tau}{l^{\ast}}\frac{\text{dl}^{\ast}}{d\tau}>0\text{ if }B=B_{4} ,$$
  $$ \frac{\tau}{\mu^{\ast}}\frac{d\mu^{\ast}}{d\tau} \! >\!0\text{ if (} B = B_{3}\text{ and }d\!>\!-\frac{\varepsilon_{\text{cB}}}{\varepsilon_{\text{cc}}}\text{) or (}B = B_{4}\text{ and }d\!<\!-\frac{\varepsilon_{cB}}{\varepsilon_{cc}}\text{)}, $$

• (2) if d > 1,

  $$ \begin{array}{@{}rcl@{}} \frac{\tau}{B_{5}}\frac{\text{dB}_{5}}{d\tau} & >&0,\\ \frac{\tau}{l^{\ast}}\frac{\text{dl}^{\ast}}{d\tau} & >&0,\\ \frac{\tau}{\mu^{\ast}}\frac{d\mu^{\ast}}{d\tau} & >&0\text{ if } d<-\frac{\varepsilon_{\text{cB}}}{\varepsilon_{\text{cc}}}\text{.} \end{array} $$

Proof

See the Appendix. □

Interestingly, the critical value of the dilution effect remains 1 as in Propositions 5 and 6. Proposition 7 deserves some economic interpretations.

The negative impact of τ on k^∗ is far from surprising. Indeed, since τ is levied on the production level, a higher green tax rate reduces production and income, entailing a lower capital level at the end. Conversely, the effect of τ on biodiversity is ambiguous and depends on two key elements: the shape of the reproduction function and the magnitude of the dilution effect. First, let us consider the case of a low dilution effect (0 < d < 1) and remark that, since a higher green tax always reduces the capital level, it also lowers the right-hand side of Eq. 29. That is, if at the steady state, the economy is located along the upward-sloping branch of φ (that is B = B₃), a higher green tax always lowers the biodiversity level. Conversely, if the economy is located on the downward-sloping branch of φ (that is B = B₄), a higher tax level lowers the right-hand side of Eq. 29, and hence, increases the biodiversity level at the steady state. The ambiguous effect of the green tax on the biodiversity reveals that the shape of the reproduction function matters when the dilution effect is low (0 < d < 1). Furthermore, the fact that a higher green tax rate impairs the biodiversity level (when $B<\bar {B}$) is counter–intuitive and refers to the static green paradox introduced in [6, 10]).

The impact of the green tax on the labor supply mimics that on the biodiversity level because of Assumption 1. Indeed, because of the dilution effect, a lower biodiversity level implies a more infective disease (a higher β jointly with a lower γ) which reduces the labor supply (l = γ/β).

Now, consider the case of a higher dilution effect (d > 1). In this case, φ is always a decreasing function of B (see the proof of Proposition 6). Hence, a higher tax rate lowers the right-hand side of Eq. 29 which always leads to a higher biodiversity level.

Summing up, we observe that the green paradox (that is the negative impact of the green tax on the biodiversity level) only occurs for low levels of biodiversity ($B<\bar {B}$) and dilution effect (0 < d < 1).

It is worthy to notice that when the biodiversity becomes a prominent determinant of human health because of a sufficiently large dilution effect (d > 1), the unpleasant (static) green paradox is ruled out. Moreover, in the previous section, we have observed that the complementarity between economic activities (production) and nature entails a biodiversity preservation in the long run through a large dilution effect. It follows that a large dilution effect (d > 1) has a double benefit: (1) it preserves the biodiversity in the long run and (2) it rules out the green paradox.

μ is a shadow price (marginal utility of consumption: $\mu =u_{c}\left (c,B\right )$). From an economic point of view, it is more interesting to characterize the impact of the green tax on consumption than on this unobservable variable.

We observe that

$$ c^{\ast}\left( \tau\right) =\frac{\theta+\left[ 1-\alpha\left( k^{\ast }\left( \tau\right) \right) \right] \delta}{\alpha\left( k^{\ast}\left( \tau\right) \right) }k^{\ast}\left( \tau\right) l^{\ast}\left( \tau\right) \text{.} $$

(36)

In order to provide a clear-cut comparative statics, we focus on the Cobb–Douglas case. In this case, the capital share $\alpha \left (k^{\ast }\right )$ becomes a constant and Eq. 36 entails the following:

$$ \frac{\tau}{c^{\ast}}\frac{\partial c^{\ast}}{\partial\tau}=\frac{\tau}{k^{\ast}}\frac{\partial k^{\ast}}{\partial\tau}+\frac{\tau}{l^{\ast}} \frac{\partial l^{\ast}}{\partial\tau}\text{.} $$

(37)

Therefore, the impact of the green tax on consumption can be disentangled in its effects on the production factors.

Proposition 8

Consider the endemic steady state l = γ/β with a = b and σ = 1 (Cobb–Douglas technology).

• (1) If 0 < d < 1 (low dilution effect),

  $$ \frac{\partial c^{\ast}}{\partial\tau}>0\text{ iff }B_{4}<\frac{1}{2}\text{.} $$

• (2) If d > 1 (high dilution effect),

  $$ \frac{\partial c^{\ast}}{\partial\tau}>0\text{ iff }B_{5}<1/2\text{.} $$

Proof

See the Appendix. □

The last proposition shows that the effect of τ on the consumption demand is ambiguous in the long run. This deserves some economic interpretations.

Consider the case of a low dilution effect (0 < d < 1) and suppose, for simplicity, no capital depreciation. As seen before, a higher green tax rate always lowers the capital level (see Eq. 35). In addition, since the economy is located along the increasing branch of the reproduction function (B = B₃), this implies a drop in the biodiversity level, rendering the disease more infective, which lowers the labor supply. Focus on expression (24): $c^{\ast }=l^{\ast }\left (1-\tau \right ) f\left (k^{\ast }\right )$. Since τ increases and k^∗ and l^∗ decrease, c^∗ decreases.

Now, assume that the economy is located along the decreasing branch of the reproduction function (B = B₄). In this case, a higher green tax rate implies more biodiversity making the disease less infective (dilution effect) and raising the labor supply. In contrast, as above, the capital intensity lowers. Then, l^∗ increases, while $\left (1-\tau \right ) f\left (k^{\ast }\right )$ decreases. The total impact of τ on $c^{\ast }=l^{\ast }\left (1-\tau \right ) f\left (k^{\ast }\right )$ is ambiguous. In order to know whether the increase in the labor supply dominates the decrease of $\left (1-\tau \right ) f\left (k^{\ast }\right )$, we have to focus on the elasticity (32) of the reproduction function φ. B = B₄ jointly with 0 < d < 1 implies $\varphi ^{\prime }\left (B\right ) <0$. The slope of φ becomes flatter when B₄ < 1/2 and steeper when B₄ > 1/2. In other words, an increase in the green tax rate has a larger effect on biodiversity and labor supply when B₄ < 1/2. Therefore, since $B_{4} >\bar {B}$, if B₄ < 1/2, the increase in labor supply l^∗ dominates the drop of $\left (1-\tau \right ) f\left (k^{\ast }\right )$ which implies a higher consumption level in the long run. Conversely, if B₄ > 1/2, the increase in labor income is dominated and consumption lowers in the long run.

Similar interpretations hold in the case of a strong dilution effect (d > 1).

5 Local Dynamics Around the Endemic Steady State

As seen above, the disease-free regime is the same as the one considered by Bosi and Desmarchelier [10]. Then, the novelty is the study of the endemic regime since Bosi and Desmarchelier [10] do not consider the spread of an infectious disease. In their paper, the authors point out that a limit cycle (Hopf bifurcation) can occur around the steady state with the highest biodiversity level (denoted by B₂ in the current paper) since biodiversity affects the marginal utility of consumption. As seen above, a kind of relationship seems to exist between the disease-free and the endemic regimes when the dilution effect is low (d < 1). More precisely, in these two cases, under a low environmental impact of production, two steady states exist (Propositions 4 and 5): the one with a low biodiversity level (B₁ and B₃ respectively for the disease-free regime and for the endemic one) and the other with a high biodiversity level (B₂ and B₄ respectively for the disease-free regime and for the endemic one). Moreover, when the environmental impact of production becomes excessive, we have observed that the two steady states collide and disappear (B₁ and B₂ for the disease-free regime and B₃ and B₄ for the endemic regime). Because of this relationship, we expect to recover around B₄ the limit cycle surrounding B₂ pointed out by Bosi and Desmarchelier [10]. Nevertheless, we have observed earlier that the situation becomes very different under a large dilution effect (d > 1). Indeed, the complementarity between production activities and biodiversity induced by a strong dilution effect ensures that there exist always a unique positive steady state (Proposition 6): biodiversity is always preserved in the long run.

Therefore, two questions arise: (1) is the limit cycle surrounding B₂ (see [10]) preserved around B₄?; (2) is the complementarity between production activities and biodiversity induced by a large dilution effect able to prevent the existence of a limit cycle around B₅? The current section will focus on these two questions.

To study the equilibrium transition, we linearize the dynamic system (19)–(22) around the endemic steady state l = l^∗. Noticing that $\omega \left (k\right ) /\left [ k\rho \left (k\right ) \right ] =\left [ 1-\alpha \left (k\right ) \right ] /\alpha \left (k\right )$, we get the Jacobian matrix as follows:

$$ J=\left[ \begin{array}{cccc} 0 & n\frac{\mu^{\ast}}{k^{\ast}} & 0 & 0\\ -\frac{q}{\varepsilon_{\mathit{cc}}}\frac{k^{\ast}}{\mu^{\ast}} & \theta & \left( m+q\right) \frac{k^{\ast}}{l^{\ast}} & \frac{k^{\ast}}{B}\left( q\frac{\varepsilon_{\mathit{cB}}}{\varepsilon_{\mathit{cc}}}-\mathit{md}\right) \\ 0 & 0 & -m & \frac{l}{B}\mathit{md}\\ 0 & \alpha\frac{B}{k^{\ast}}\left( B-1\right) & \frac{B}{l^{\ast}}\left( B-1\right) & 1-2B \end{array} \right], $$

where $q\equiv \left [ \theta +\left (1-\alpha \right ) \delta \right ] /\alpha =c^{\ast }/\left (k^{\ast }l^{\ast }\right )$, $m\equiv \beta \left (1-l^{\ast }\right )$, and $n\equiv \left (\theta +\delta \right ) \left (1-\alpha \right ) /\sigma$. To study the local dynamics of this four-dimensional system, we apply the methodology developed by Bosi and Desmarchelier [7] and based on the sums of principal minors of the Jacobian. The characteristic polynomial is given by $P\left (\lambda \right ) \equiv \lambda ^{4}-T\lambda ^{3}+S_{2}\lambda ^{2}-S_{3}\lambda +D$ where

$$ \begin{array}{@{}rcl@{}} S_{1} & =&\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4} =T, \end{array} $$

(38)

$$ \begin{array}{@{}rcl@{}} S_{2} & =&\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{1}\lambda_{4}+\lambda_{2}\lambda_{3}+\lambda_{2}\lambda_{4}+\lambda_{3}\lambda_{4}, \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} S_{3} & =&\lambda_{1}\lambda_{2}\lambda_{4}+\lambda_{1}\lambda_{3}\lambda_{4}+\lambda_{2}\lambda_{3}\lambda_{4}+\lambda_{1}\lambda_{2}\lambda_{3}, \end{array} $$

(40)

$$ \begin{array}{@{}rcl@{}} S_{4} & =&\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}=D, \end{array} $$

(41)

T and D denote the trace and the determinant of J while S₂ and S₃ represent the sum of principal minors of order two and three.

In our model,

$$ \begin{array}{@{}rcl@{}} T &=& \theta-m-2B+1,\\ S_{2} &=& \left[ 2B-1+\left( 1-B\right) d\right] m+\left( 1-2B\right) \theta\\&& -\left( 1-B\right) \mathit{dm}\alpha +q\alpha\frac{\varepsilon_{\mathit{cB}}} {\varepsilon_{\mathit{cc}}}\left( 1-B\right) -m\theta+\frac{\mathit{nq}}{\varepsilon_{\mathit{cc}} },\\ S_{3} & =&\frac{1}{\varepsilon_{\mathit{cc}}}\left[ \left( 1-2B-m\right) \mathit{nq}+\left( B-1\right) \mathit{mq}\alpha\varepsilon_{\mathit{cB}}\right]\\ &&+\left( B-1\right) \mathit{dmq}\alpha+\left[ 2B-1+\left( 1-B\right) d\right] m\theta, \\ D & =&\frac{\mathit{mnq}}{\varepsilon_{\mathit{cc}}}\left[ 2B-1+\left( 1-B\right) d\right] \text{.} \end{array} $$

(42)

According to Bosi and Desmarchelier [10], a Hopf bifurcation can arise around the higher disease-free steady state B₂ because ε_cB ≠ 0. To characterize the occurrence of limit cycles in our model, we introduce, for simplicity, a isoelastic utility function as follows:

$$ u\left( c_{t},B_{t}\right) =\frac{\left( c_{t}B_{t}^{\eta}\right)^{1-\varepsilon}}{1-\varepsilon}\text{.} $$

(43)

The functional form (43) is widely used in economics.⁵1/ε > 0 is the intertemporal elasticity of substitution of the composite good $c_{t}B_{t}^{\eta }$ while η > 0 represents the weight of biodiversity in the household’s utility. Interestingly, in this case, the elasticities of preferences (8) are invariant to the steady state:

$$ \left[ \begin{array}{cc} \varepsilon_{\mathit{cc}} & \varepsilon_{\mathit{cB}}\\ \varepsilon_{\mathit{Bc}} & \varepsilon_{\mathit{BB}} \end{array} \right] =\left[ \begin{array}{cc} -\varepsilon & \eta\left( 1-\varepsilon\right) \\ 1-\varepsilon & \eta\left( 1-\varepsilon\right) -1 \end{array} \right] \text{.} $$

We observe that ε_cB > 0 (< 0) if and only if ε < 1 (> 1). Let ε₁ ≡ ε_cc = −ε < 0, $\varepsilon _{2}\equiv \varepsilon _{\mathit {cB}}=\eta \left (1-\varepsilon \right )$ and

$$ \varepsilon_{H}\equiv\varepsilon_{1}\frac{z_{3}-T\frac{Z+\sqrt{Z^{2} -4mD\left( m+T\right) }}{2\left( m+T\right) }}{m\alpha q\left( 1-B\right) }, $$

(44)

with

$$ \begin{array}{@{}rcl@{}} z_{3} \!&\! \equiv\!&\! \text{qm}\alpha d\left( B - 1\right)+\left( T - \theta\right) \text{nq}/\varepsilon_{1}+\theta D\varepsilon_{1}/\left( \text{nq}\right),\\ Z \!&\!\equiv\!&\! m\left[ m\alpha d\left( B - 1\right) +\text{nq}/\varepsilon_{1} + D\varepsilon_{1}/\left( \text{nq}\right) +\left( T - \theta\right) \theta\right] +z_{3}\text{.} \end{array} $$

The following proposition characterizes the occurrence of a Hopf bifurcation under a low (d < 1) and large (d > 1) dilution effect.

Proposition 9

Let $B<\left (1+\theta \right ) /2$. If

• (1) 0 < d < 1 (low dilution effect) and B = B₄ (large biodiversity)

• (2) d > 1 (large dilution effect)

then a limit cycle occur near the endemic steady state if and only if ε₂ = ε_H.

Proof

See the Appendix. □

Proposition 9 shows the possibility of limit cycle around the endemic steady state with the higher biodiversity (B₄) in the spirit of Bosi and Desmarchelier [10] where the limit cycle surrounds the (disease-free) steady state with the higher biodiversity (B₂). The only difference between B₂ and B₄ is the existence of a (low) dilution effect. Qualitatively, nonlinear dynamics under a low dilution effect (d < 1) look like those with no dilution effect. However, now, the limit cycle is preserved also when the dilution effect becomes large (d > 1). Even if, as seen above, a sufficiently large dilution effect entails the preservation of biodiversity in the long run, it does not shelter the bioeconomic system from the existence of (limit) cycles.

To understand the preservation limit cycles under a large dilution effect (d > 1), we have to consider the sign of ε_H. Since ε_cB = ε_H at the Hopf bifurcation point, we assume a positive cross elasticity ε_cB > 0 (complementarity between consumption and biodiversity) to avoid the unpleasant and meaningless case ε_H < 0. In the next section, to convince the reader, we will provide a numerical simulation with a limit cycle around B₄ with ε_H > 0 (that is a large elasticity of intertemporal substitution: 1/ε > 1).

Now, let the economy be at the steady state today and assume an exogenous rise in the pollution level. Equation 16 implies a drop in biodiversity with two consequences: (1) a lower labor supply ($l_{t}=\gamma \left (B_{t}\right ) /\beta \left (B_{t}\right )$ decreases: dilution effect) and (2) a lower consumption demand due to complementarity ($\mu _{t}=u_{c}\left (c_{t} ,B_{t}\right )$ decreases). The Euler equation (intertemporal consumption smoothing) implies $\dot {\mu }_{t}/\mu _{t}=\theta +\delta -\left (1-\tau \right ) \rho \left (k_{t}\right ) <0$. Thus, $\rho \left (k_{t}\right )$ increases from the MGR $\left (\theta +\delta \right ) /\left (1-\tau \right )$ today to the new transition value tomorrow and, since ρ is a decreasing function, the capital intensity k_t lowers and the average productivity $f\left (k_{t}\right )$ as well. Pollution is given by $P_{t}=\left (a-b\tau \right ) l_{t}f\left (k_{t}\right )$. Hence, under Assumption 4, the drops in labor supply l_t and in productivity $f\left (k_{t}\right )$ entail a lower pollution level, and hence a higher biodiversity level. Thus, a higher pollution today entails a weaker pollution tomorrow giving rise to an endogenous fluctuation. According to this interpretation, the dilution effect (a biodiversity loss implies less labor supply) amplifies the drop of production induced by the drop of consumption (because of the complementarity: ε_cB < 0) at the origin of the endogenous fluctuation. In other terms, a larger dilution effect magnifies the cycle instead of preventing it.

The existence of a limit cycle makes sense from a biological point of view. The biodiversity level fluctuates over time: a period with a larger biodiversity can be followed by another age with lower biodiversity and so on. This is reminiscent of the five recurrent mass extinctions experienced by planet Earth in the past 540 million years [4].

6 Simulations

The analytical study of local dynamics has revealed the existence of limit cycles under either a low or a high dilution effect. Now, let us convince the reader through a numerical illustration. This simulation will convince her not only about the occurrence of cycles but also about the existence of more sophisticated dynamics (of codimension two) such as the Bogdanov–Takens bifurcation generated by the system (19)–(22).

Since the Bogdanov–Takens bifurcation arises only under a low dilution effect, for brevity’s sake, we will focus only on the case of a low dilution effect. More precisely, we will consider numerically the dynamics around the steady state with a higher biodiversity $\left (\mu ^{\ast },k^{\ast },l^{\ast } ,B_{4}\right )$.

We reconsider the isoelastic utility (43) and the isoelastic functions (30). For simplicity, we use also a Cobb–Douglas production function: $f\left (k_{t}\right ) =Ak_{t}^{\alpha }$. According to the MGR (23), we find the stationary capital level:

$$ k^{\ast}=\left[ \frac{\alpha A\left( 1-\tau\right) }{\theta+\delta}\right]^{\frac{1}{1-\alpha}}\text{.} $$

Replacing k^∗ in Eq. 29, we obtain the biodiversity level as solution of the following:

$$ \left( 1-B\right) B^{1-d}=A\left( a-b\tau\right) \frac{A_{\gamma} }{A_{\beta}}k^{\ast\alpha}\text{.} $$

(45)

According to Proposition 5, Eq. 45 possesses two solutions if and only if

$$ a<\bar{a}=b\tau+\frac{A_{\beta}}{A_{\gamma}}\frac{\left( 1-\bar{B}\right) \bar{B}^{1-d}}{\text{Ak}^{\ast\alpha}}, $$

where d ≡ ε_γ − ε_β > 0.

Consider now the calibration in Table 1.

Table 1 Parameter values

α, δ, and 𝜃 are set at their usual quarterly values⁶ while b and τ⁷ are fixed as in [8]. According to Table 1, we observe that ε_H > 0 since ε < 1.⁸ That is, we have fixed ε to ensure that ε < 1. In addition, as explaining before, we focus on the low dilution effect case (d < 1). By fixing ε_γ = −ε_β = 0.25, we ensure that d ≡ ε_γ − ε_β = 1/2 < 1. Finally, since there are no critical values concerning the scaling parameters A, A_β, and A_γ for the existence of the Hopf bifurcation (see Proposition 9), we choose to fix each of them equal to the unity.

The calibration provided in Table 1 yields $\bar {a}=0.1276$. We fix $a=0.127<\bar {a}$ and we solve (45) for B. As expected (see Proposition 5), we obtain two roots: B₃ = 0.2968 and B₄ = 0.3713 (see Proposition 5).We observe that the necessary (but not sufficient) condition for the occurrence of a Hopf bifurcation in Proposition 9 is satisfied: $\bar {B}=0.333<B_{4}=0.3713<\left (1+\theta \right ) /2= 0.505$.

Focusing on B₄ = 0.3713, we compute η such that $\varepsilon _{2} =\eta \left (1-\varepsilon \right ) =\varepsilon _{H}$. We obtain η_H = 0.27569 > 0. Therefore, under this calibration (Table 1), when a = 0.127, the system undergoes a Hopf bifurcation at η_H and experiences a limit cycle near B₄ = 0.3713.

Summing up, under the calibration of Table 1, we get a Hopf bifurcation at a = 0.127 and a saddle-node bifurcation at a = 0.1276 near the higher endemic steady state (B = B₄ with l = γ/β and 0 < d < 1).

After having seen how to calibrate the model to find a Hopf bifurcation, we deepen our approach considering an equilibrium continuation.⁹ We aim to plot the Hopf bifurcation curve and the saddle-node bifurcation curve in the $\left (a,\eta \right )$-plane and to show the occurrence of the Bogdanov–Takens bifurcation when these bifurcation curves meet each others. We will refer to Fig. 2 where LP, H, BT, and GH stand for limit point (elementary saddle-node), Hopf, Bogdanov–Takens and generalized Hopf. These points are computed and represented by MATCONT when the corresponding bifurcations occur near the steady state.

Fig. 2

The equilibrium continuation

To perform the equilibrium continuation using MATCONT, we consider first the bifurcation of codimension one (Hopf and the saddle node). We fix η = 0.27569 as above and we set an arbitrary value for a at which no bifurcation occurs near the endemic steady state: a = 0.1268 ≡ a₀. In this case, according to Table 1, the endemic steady state becomes the following:

$$ \left( \mu^{\ast},k^{\ast},l^{\ast},B_{4}\right) =\left( 0.7354,28.385671,0.61423845,0.37728887\right) \text{.} $$

Let MATCONT raise a from a₀ ≡ 0.1268 to $\bar {a}=0.1276$ keeping η = 0.27569 as constant. In Fig. 2, we are moving to the right along the horizontal line HLP. MATCONT detects a Hopf bifurcation (H) at a = a_H = 0.127 and a saddle-node bifurcation (LP) at $a=\bar {a}$.

Now, let us move from H to BT along the locus of all the Hopf bifurcations: for any a, there is a Hopf critical value as follows:

$$ \eta_{H}\left( a\right) \equiv-\frac{\varepsilon}{1-\varepsilon}\frac {z_{3}\left( a\right) -T\left( a\right) \frac{Z\left( a\right) +\sqrt{Z\left( a\right)^{2}-4m\left( a\right) D\left( a\right) \left[ m\left( a\right) +T\left( a\right) \right] }}{2\left[ m\left( a\right) +T\left( a\right) \right] }}{m\left( a\right) \alpha q\left[ 1-B\left( a\right) \right] }\text{.} $$

The Hopf bifurcation curve $\left \{ \left (a,\eta _{H}\left (a\right )\right ) \right \}$ is precisely represented in Fig. 2 by the curve HBT.

For any η, the elementary saddle-node bifurcation value for a is $\bar {a}=0.1276$ (the line LPBT is vertical because $\bar {a}$ does not depend on η). In particular, the limit point corresponding to η = 0.27569 is $\text {LP}=\left (0.1276,0.275 69\right )$. The vertical line LPBT represents a third equilibrium continuation, the set of all the pairs $\left (a,\eta \right ) =\left (\bar {a},\eta \right )$ for which a saddle-node bifurcation occurs.

To sum up, increasing a from a₀ = 0.1268 to $\bar {a}=0.1276$, we obtain all the Hopf bifurcations along the curve $\text {HBT}\equiv \left \{ \left (a,\eta _{H}\left (a\right ) \right ) \right \}_{a\in \left [ a_{0},\bar {a}\right ] }$ going from H to BT. Moreover, in the range $\left [ a_{0},\bar {a}\right ) \ni a$, we find two distinct steady states. When a attains the maximal value $\bar {a}$, these two steady states coalesce and the Hopf bifurcation point $\left (a,\eta _{H}\left (a\right ) \right )$ reaches the ending point BT along the curve HBT while the economy experiences a Bogdanov–Takens bifurcation. Indeed, a Bogdanov–Takens bifurcation generically arises when a Hopf bifurcation curve crosses a locus of saddle-node bifurcations.

Along the locus of Hopf bifurcations, two generalized Hopf (Bautin) bifurcations also appear. A Hopf bifurcation can be subcritical or supercritical, leading respectively to an unstable or stable limit cycle. The generalized Hopf bifurcation point implies a change in the stability of the limit cycle arising near the steady state, that is, the bifurcation from subcritical becomes supercritical or viceversa. If the first Lyapunov coefficient (l₁) is negative (positive), the bifurcation is said to be supercritical (subcritical), leading to a stable (unstable) limit cycle near the steady state. At the generalized Hopf bifurcation point, l₁ vanishes.

Let us explain the relation between a double-Hopf and a generalized Hopf bifurcation.¹⁰ At the double-Hopf bifurcation, two limit cycles emerge simultaneously. The interaction between these two limit cycles can produce a wide range of dynamics depending upon higher order terms of the Taylor series, such as a torus or local chaos. Concerning the Jacobian matrix, a double pair of purely imaginary eigenvalues appears at the double-Hopf bifurcation. In the case of a generalized Hopf bifurcation, the arising limit cycle is unique, as for a standard Hopf bifurcation: the Jacobian possesses a single pair of purely imaginary eigenvalues. The distinction between a standard or a generalized Hopf bifurcation rests on the value of the first-order Lyapunov coefficient and, thus, on the higher-order terms of the Taylor representation of the dynamical system. Indeed, at the generalized Hopf bifurcation, the first Lyapunov coefficient is equal to zero which means a change of stability for the limit cycle.

At the Hopf bifurcation point (H), the steady state is given by the following:

$$ \left( \mu^{\ast},k^{\ast},l^{\ast},B_{4}\right) =\left( 0.736723,28.38567,0.609338,0.371292\right), $$

with eigenvalues:

$$ \lambda_{1}=-~0.34187\text{, }\lambda_{2}=0.108813,\text{ and }\lambda_{3}=0.0539646i=-\lambda_{4}\text{.} $$

In order to visualize the limit cycle arising at the Hopf bifurcation point, we project the four-dimensional dynamics on a three-dimensional space. Since the shadow price μ is not a directly observable variable, we prefer to represent the trajectory in the $\left (k_{t},l_{t},B_{t}\right )$ space.

The corresponding first Lyapunov coefficient is given by l₁ = 6.182045 ∗ 10^− 5 > 0. Its positivity means that the Hopf bifurcation is subcritical, that is, the limit cycle arising near the steady state is unstable (Fig. 3).

Fig. 3

The unstable limit cycle

At the saddle-node bifurcation (LP), the steady state becomes the following:

$$ \left( \mu^{\ast},k^{\ast},l^{\ast},B_{4}\right) =\left( 0.745 69,28.38567,0.577347,0.333333\right), $$

with eigenvalues:

$$ \lambda_{1}=-~0.401245\text{, }\lambda_{2}=0\text{, }\lambda_{3} =0.0204607,\text{ and }\lambda_{4}=0.16789\text{.} $$

At the Bogdanov–Takens bifurcation (BT), when $a=\bar {a}=0.1276$ jointly with η = η_BT = 0.414576, the steady state becomes the following:

$$ \left( \mu^{\ast},k^{\ast},l^{\ast},B_{4}\right) =\left( 0.690915,28.385671,0.577350,0.333333\right), $$

with eigenvalues:

$$ \lambda_{1}=-0.399 61\text{, }\lambda_{2}=\lambda_{3}=0,\text{ and } \lambda_{4}=0.186 71\text{.} $$

The Bogdanov–Takens bifurcation occurs when conditions for the elementary saddle-node bifurcation and for the Hopf bifurcation meet each other.

As in Kuznetsov et al. [22], at the Bogdanov–Takens point, the orbit describes a parasitic loop near the saddle-point (Fig. 4). The parasitic loop typically arises when the limit cycle and the saddle-point coalesce.

Fig. 4

The parasitic loop

As above, to represent the trajectory, we project the four-dimensional dynamics on the three-dimensional $\left (\mu _{t},l_{t},B_{t}\right )$ space, where the parasitic loop appears (Fig. 4).

At the generalized Hopf bifurcations (GH), we obtain the following:

Parameters	Steady state	Eigenvalues	l2
a = 0.12700794	μ∗ = 0.73602393	λ1 = − 0.342374	2.61837 ∗ 10− 3
η = 0.2778149	k∗ = 28.38567	λ2 = 0.109537
	l∗ = 0.60905296	λ3 = − 0.0535261i
	B4 = 0.37094551	λ4 = 0.0535261i
a = 0.12718298	μ∗ = 0.7251378	λ1 = − 0.351468	1.744669 ∗ 10− 2
η = 0.31106998	k∗ = 28.38567	λ2 = 0.122331
	l∗ = 0.60395327	λ3 = − 0.0460886i
	B4 = 0.36475955	λ4 = 0.0460886i

A GH bifurcation implies a change in the stability of the limit cycle arising through the Hopf bifurcation. Typically, such a bifurcation occurs when the first Lyapunov coefficient vanishes. This phenomenon can not be detected through a simple analysis of the eigenvalues. According to [21], a GH bifurcation is non-degenerated bifurcation if the second-order Lyapunov coefficient is different from zero (l₂ ≠ 0). It is the case under our calibration for both the GH bifurcations.

7 Conclusion

We have provided a unified framework at the crossroad of economics, ecology, and epidemiology, and studied how the negative relation between biodiversity and disease transmission (the so-called dilution effect) affects the economy in the long and the short run. More precisely, we have embedded a SIS model into a Ramsey model where a pollution externality coming from production impairs a biodiversity measure. For the sake of simplicity, we have assimilated biodiversity to a renewable resource and introduced a two-sided dilution effect assuming that both the probability to become ill and the recovery rate from the infectious disease depend on the biodiversity level. To complete the model, we have considered a proportional tax levied on production at the firm level to finance depollution.

In the long run, we have recovered a standard feature of the SIS model: a disease-free regime coexists with an endemic one. In the endemic case, the number of steady states depends on both the magnitude of the dilution effect and on the environmental impact of production. More precisely, under a low dilution effect, two cases are possible: (1) two steady states coexist if the environmental impact of production is low (the one with low biodiversity and the other with high biodiversity); (2) there is no steady state if the environmental impact of production is large enough. In case (2), biodiversity is not preserved in the long run because of the excessive human pressure on nature. When the dilution effect is high, the bioeconomic system works very differently: there is always a unique steady state whatever the environmental impact of production. In this case, the complementarity between economy and biodiversity induced by a strong dilution effect prevents human pressure from becoming lethal for nature. In addition, we have highlighted a kind of green paradox in the endemic regime: under a low dilution effect, a higher green tax rate always impairs biodiversity at the low steady state. This counter–intuitive result is comparable to the static green paradox considered in [6]. Conversely, the green paradox is over under a large dilution effect.

In the short run, limit cycles can arise under both the low and the high dilution effect through a Hopf bifurcation near the steady state. This happens in the endemic case when preferences exhibit a complementarity between biodiversity and consumption.

To sum up, a large dilution effect seems to imply a double benefit: (1) it preserves biodiversity in the long run and (2) it prevents the economy from the green paradox. However, a large dilution effect cannot shelter the economy from unpleasant biodiversity fluctuations shaped as a limit cycle around the steady state when the household’s preferences exhibit complementarity between biodiversity and consumption.

Appendix

Proof

of Proposition 2

The agent maximizes the intertemporal utility function (9) under the budget constraint (7). Setting the Hamiltonian $H_{t}=e^{-\theta t}u\left (c_{t},B_{t}\right ) +\lambda _{t}\left [ \left (r_{t} -\delta \right ) h_{t}+\bar {\omega }_{t}-c_{t}\right ]$, deriving the first-order conditions ∂H_t/∂c_t = 0, $\partial H_{t}/\partial h_{t}=-\dot {\lambda }_{t}$ and $\partial H_{t}/\partial \mu _{t}=\dot {h}_{t}$, and defining μ_t ≡ λ_te^𝜃t, we get Eqs. 10, 11, and 12. □

Proof

of Proposition 3

Consider Eqs. 4, 14, 18, and Proposition 2. □

Proof

of Proposition 7

We differentiate system (19)–(22) to find the following:

$$ \begin{array}{l} \left[ \begin{array}{c} \frac{\tau}{\mu^{\ast}}\frac{d\mu^{\ast}}{d\tau}\\ \frac{\tau}{k^{\ast}}\frac{\text{dk}^{\ast}}{d\tau}\\ \frac{\tau}{l^{\ast}}\frac{\text{dl}^{\ast}}{d\tau}\\ \frac{\tau}{B}\frac{\text{dB}}{d\tau} \end{array} \right] \mathbf{=}\left[ \begin{array}{cccc} 0 & \frac{1-\alpha}{\sigma} & 0 & 0\\ -\frac{\phi}{\varepsilon_{\text{cc}}} & \theta & \phi+\beta\left( 1-l\right) & \phi\frac{\varepsilon_{\text{cB}}}{\varepsilon_{\text{cc}}}-\gamma d\frac{1-l}{l}\\ 0 & 0 & 1 & -d\\ 0 & \alpha & 1 & -\frac{1-2B}{1-B} \end{array}\right]^{-1}\left[ \begin{array}{c} -\frac{\tau}{1-\tau}\\ \frac{\theta+\delta}{\alpha}\frac{\tau}{1-\tau}\\ 0\\ \frac{b\tau}{a-b\tau} \end{array} \right],\\\\ \text{where}\\ \phi\equiv\frac{\theta+\left( 1-\alpha\right) \delta}{\alpha}=\frac{c^{\ast }}{k^{\ast}l^{\ast}},\\ \text{That is,} \\ \left[ \begin{array}{c} \frac{\tau}{\mu^{\ast}}\frac{d\mu^{\ast}}{d\tau}\\ \frac{\tau}{k^{\ast}}\frac{\text{dk}^{\ast}}{d\tau}\\ \frac{\tau}{l^{\ast}}\frac{\text{dl}^{\ast}}{d\tau}\\ \frac{\tau}{B}\frac{\text{dB}}{d\tau} \end{array} \right] =\left[ \begin{array}{c} \varepsilon_{\text{cc}}\left[ \left( \frac{b\tau}{a-b\tau}+\frac{\alpha\sigma }{1-\alpha}\frac{\tau}{1-\tau}\right) \frac{1-B}{\bar{B}-B}\dfrac {d+\frac{\varepsilon_{\text{cB}}}{\varepsilon_{\text{cc}}}}{d-2}-\left[ 1+\frac{\alpha }{1-\alpha}\frac{\sigma\theta+\left( 1-\alpha\right) \delta}{\theta+\left( 1-\alpha\right) \delta}\right] \frac{\tau}{1-\tau}\right] \\ -\frac{\sigma}{1-\alpha}\frac{\tau}{1-\tau}\\ \left( \frac{\alpha\sigma}{1-\alpha}\frac{\tau}{1-\tau}+\frac{b\tau}{a-b\tau }\right) \frac{1-B}{\bar{B}-B}\frac{d}{d-2}\\ \left( \frac{\alpha\sigma}{1-\alpha}\frac{\tau}{1-\tau}+\frac{b\tau}{a-b\tau }\right) \frac{1-B}{\bar{B}-B}\frac{1}{d-2} \end{array} \right] \text{.} \\\\\\\\\\\\\\\\\\\\\\\\\\ \end{array} $$

(46)

Under Assumption 4 and $B\in \left (0,1\right )$, we obtain easily Proposition 7. □

Proof

of Proposition 8

In the Cobb–Douglas case, σ = 1 and, according to expressions (46), (37) yields the following:

$$ \frac{\tau}{c^{\ast}}\frac{\partial c^{\ast}}{\partial\tau}=-\frac{1} {1-\alpha}\frac{\tau}{1-\tau}+\left( \frac{\alpha}{1-\alpha}\frac{\tau }{1-\tau}+\frac{b\tau}{a-b\tau}\right) \frac{1-B}{\bar{B}-B}\frac{d} {d-2}\text{.} $$

In the case a = b, we obtain the following:

$$ \begin{array}{@{}rcl@{}} \frac{\tau}{c^{\ast}}\frac{\partial c^{\ast}}{\partial\tau}&=&\frac{1}{1-\alpha }\frac{\tau}{1-\tau}\left( \frac{1-B}{\bar{B}-B}\frac{d}{d-2}-1\right)\\ &=&\frac{1}{1-\alpha}\frac{\tau}{1-\tau}\frac{1-2B}{\left( d-2\right) \left( \bar{B}-B\right) }\text{.} \end{array} $$

Proposition 8 immediately follows. □

Proof

of Proposition 9

According to Corollary 15 of [7], a Hopf bifurcation arises iff S₂ = S₃/T + DT/S₃ and T and S₃ have the same sign.

Let us rewrite S₂ and S₃ as follows:

$$ \begin{array}{@{}rcl@{}} S_{2} & =&\frac{Z}{m}-\frac{T}{m}\frac{S_{3}}{T}, \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} S_{3} & =&z_{3}-m\alpha q\left( 1-B\right) \frac{\varepsilon_{2} }{\varepsilon_{1}}\text{.} \end{array} $$

(48)

Replacing (47), equation

$$ S_{2}=\frac{S_{3}}{T}+D\frac{T}{S_{3}}, $$

becomes the following:

$$ \frac{S_{3}}{T}=\frac{Z\pm\sqrt{Z^{2}-4mD\left( m+T\right) }}{2\left( m+T\right) }\text{.} $$

(49)

We observe that, if 0 < d < 1, D < 0 < m + T iff $\bar {B}<B<\left (1+\theta \right ) /2$. In this case, m + T > 0 and $4mD\left (m+T\right ) <0$. If d > 1, then D < 0 and, thus, D < 0 < m + T iff $B<\left (1+\theta \right ) /2$. Even in this case, m + T > 0 and $4mD\left (m+T\right ) <0$.

Then, in both the cases,

$$ \left( \frac{S_{3}}{T}\right)_{-}<0<\left( \frac{S_{3}}{T}\right)_{+}\text{.} $$

Clearly, the solution ε₂ of

$$ \frac{S_{3}\left( \varepsilon_{2}\right) }{T}=\left( \frac{S_{3}} {T}\right)_{-}<0, $$

is not acceptable as Hopf bifurcation value because T and S₃ have opposite sign. Let ε_H be solution of the following:

$$ \frac{S_{3}\left( \varepsilon_{2}\right) }{T}=\left( \frac{S_{3}} {T}\right)_{+}\text{.} $$

Replacing Eq. 48 in the LHS and Eq. 49 in the RHS, we obtain Eq. 44. □