Emmanuelle Augeraud-Véron
email: emmanuelle.augeraud@u-bordeaux.fr

Giorgio Fabbri
email: giorgio.fabbri@univ-grenoble-alpes.fr

Katheline Schubert
email: schubert@univ-paris1.fr

PREVENTION AND MITIGATION OF EPIDEMICS: BIODIVERSITY CONSERVATION AND CONFINEMENT POLICIES

Keywords: Biodiversity, COVID-19, prevention, mitigation, epidemics, Poisson processes, recursive preferences Q56, Q57, Q58, O13, C61

This paper presents a first model integrating the relation between biodiversity loss and zoonotic pandemic risks in a general equilibrium dynamic economic set-up. The occurrence of pandemics is modeled as Poissonian leaps in economic variables. The planner can intervene in the economic and epidemiological dynamics in two ways: first (prevention), by deciding to conserve a greater quantity of biodiversity to decrease the probability of a pandemic occurring, and second (mitigation), by reducing the death toll through a lockdown policy, with the collateral effect of affecting negatively labor productivity. The policy is evaluated using a social welfare function embodying society's risk aversion, aversion to fluctuations, degree of impatience and altruism towards future generations. The model is explicitly solved and the optimal policy described. The dependence of the optimal policy on natural, productivity and preference parameters is discussed. In particular the optimal lockdown is more severe in societies valuing more human life, and the optimal biodiversity conservation is larger for more "forward looking" societies, with a small discount rate and a high degree of altruism towards future generations. Moreover, societies accepting a large welfare loss to mitigate the pandemics are also societies doing a lot of prevention. After calibrating the model with COVID-19 pandemic data we compare the mitigation efforts predicted by the model with those of the recent literature and we study the optimal prevention-mitigation policy mix.

Introduction

The hopes of the post-war period that infectious diseases were behind us thanks to control and treatment improvements [START_REF] Fisher | The plague makers[END_REF] have proved to be false: the number of emerging infectious diseases (EID) has continued to rise since the 1950s [START_REF] Smith | Global rise in human infectious disease outbreaks[END_REF]. The fear of a pandemic has remained vivid in the scientific community, as shown by the Clade X exercise 1 hosted at the Johns Hopkins Center for Health Security in 2018. Indeed, the number of outbreaks leading to epidemics and even pandemics has accelerated sharply over the last twenty years.

60% of these EID are caused by zoonotic pathogens, mainly (72%) of wildlife origin [START_REF] Jones | Global Trends in Emerging Infectious Diseases[END_REF]. Examples include AIDS, SARS (Severe Acute Respiratory Syndrom), MERS (Middle East Respiratory Syndrom), Nipah Virus, Avian influenza, Ebola virus and Influenza A virus subtype H1N1, as well as COVID-19. The large number of EID of zoonotic origin makes it natural to link the emergence of these diseases to the loss of biodiversity. The current COVID-19 pandemic, caused by the transmission of a pathogen from animals to humans, has put this link to the forefront of the stage. While all the underlying mechanisms are not fully understood, a growing scientific literature documents this complex link [START_REF] Morand | Infectious diseases and their outbreaks in Asia-Pacific: biodiversity and its regulation loss matter[END_REF][START_REF] Morand | Biodiversity and disease transmission[END_REF]. Both a dilution effect (decreasing relationship between biodiversity and EIDs, [START_REF] Civitello | Biodiversity inhibits parasites: broad evidence for the dilution effect[END_REF] and an amplification effect (increasing relationship between biodiversity and EIDs, [START_REF] Wood | Human infectious disease burdens decrease with urbanization but not with biodiversityPhil[END_REF] are at work simultaneously [START_REF] Rohr | Towards common ground in the biodiversity-disease debate[END_REF]. The predominance of one effect over the other depends on the spatial scale at which the phenomenon is studied [START_REF] Johnson | Frontiers in research on biodiversity and disease[END_REF]. On a national or global scale, the dilution effect dominates [START_REF] Halliday | Biodiversity loss underlies the dilution effect of biodiversity[END_REF][START_REF] Morand | Infectious diseases and their outbreaks in Asia-Pacific: biodiversity and its regulation loss matter[END_REF] and conserving biodiversity appears as a prevention against EIDs.

There are at least two reasons for the dilution effect [START_REF] Keesing | Impacts of biodiversity on the emergence and transmission of infectious diseases[END_REF]. Firstly, the decline in biodiversity leads to an increase in the prevalence and transmission rates at the local level and to a selection effect of the most harmful pathogenic strains. Second, habitat destruction brings species together and brings them closer to humans [START_REF] Wolfe | Bushmeat hunting, deforestation, and prediction of zoonoses emergence[END_REF]. 2 The promiscuity between several species, in the wild, in captivity or in breeding, increases the risk of transmission and mutation of pathogens, and makes transmission to humans more likely [START_REF] Logiudice | The ecology of infectious disease: effects of host diversity and community composition on Lyme disease risk[END_REF].

The economic cost of these diseases is very high [START_REF] Sands | Assessment of economic vulnerability to infectious disease crises[END_REF]. The West African Ebola epidemic of 2014 led to a 10% loss of GDP in Sierra Leone and Guinea (World Bank, 2014). The COVID-19 pandemic and the policy measures adopted to fight it, that, in the absence of a vaccine or effective treatment, led to the containment of more than half of the planet during more than 2 months, resulted in severe recessions in most countries, probably the worst since the Great Depression. In the Interim Report of September 2020 of the OECD Economic Outlook (OECD, 2020) global growth is projected to -4.5% in 2020, and growth in the Euro area to -7.9%, unusually large uncertainties around these figures being acknowledged. The economic costs are direct medical costs and mostly indirect costs coming from the disruption of supply, in the industry, in the transportation sector and particularly in aviation, in the arts and entertainment sector, in the tourism sector etc.

The basic question we are interested in is whether governments, at the world level, should coordinate to invest in biodiversity conservation as a prevention policy against the risk of emerging infectious diseases, or whether they should rely on mitigation policies reducing social interactions like lockdowns once the risk materializes, or also whether they should do both. There are two levels to this question. First, a choice has to be made between investing upstream (and how much) in biodiversity conservation, at the price of an economic loss, to reduce the probability of occurrence of pandemics, or not. Second, once a pandemic hits, the severity of the mitigation policy put in place to attenuate its negative effects on health and mortality has to be decided. We design a stylized model allowing us to shed light on these issues, first from a theoretical point of view, and then with a calibration of the model to data on COVID-19 and simulations. A huge literature in economics on the COVID-19 pandemic has emerged since March 2020. 3 But the question of the links between biodiversity losses and EIDs we study here are, to the best of our knowledge, totally overlooked in this economic literature.

The theoretical framework we adopt is the one of a long-term macrodynamic economic model embedding biodiversity on the one hand, and epidemics, through a standard SIR model, on the other hand. The planner decides on the effort to be made to preserve biodiversity in order to reduce the likelihood of new EID. How much biodiversity is preserved depends on the size of land left undeveloped 4 , that supports intact natural ecosystems such as primary forests. This size impacts the probability (and then the 3 As an example, [START_REF] Brodeur | A Literature Review of the Economics of COVID-19[END_REF] document that between March 2020 and the end of May 2020 there had been 106 NBER working papers related to The relation between the area of wild land and the number of species it contains has been known since the seminal works of [START_REF] Arrhenius | Species and Area[END_REF] and [START_REF] Gleason | On the relation between species and area[END_REF]. Recent results allow to improve the parametrization of the dependence, see Section 2 for a discussion. frequency) of occurrence of epidemic outbreaks. It comes at an economic cost, as land used for human activities (agricultural production, infrastructures, human settlements etc.) is a direct input in the production process.

In addition to this risk-reducing policy, the planner also has the possibility to decide on lockdown mitigation policies in the event of a pandemic, in order to reduce population mortality, at the cost of a reduction in productive activities.

Given the simplicity of the model we can solve it explicitly. Using a social welfare function embodying society's risk aversion, aversion to fluctuations, degree of impatience and altruism towards future generations, we characterize the optimal mitigation policy and the optimal allocation of land to biodiversity conservation and study their behaviors in terms of the model's parameters.

We show that the optimal lockdown is more severe in societies valuing more human life, and that the optimal biodiversity conservation is greater in more "forward looking" societies, with a small discount rate and a high degree of altruism towards future generations. We also show that societies accepting a large welfare loss to mitigate the pandemics are also societies doing a lot of prevention, not to have to incur the loss too often and so all the more since risk aversion or the risk of pandemics absent any biodiversity are higher.

To calibrate the model we use data from [START_REF] Gollier | Cost-benefit analysis of age-specific deconfinement strategies[END_REF] on the consequences in terms of mortality and economic activity of the laissez-faire, a "flatten the curve" mitigation strategy and a "crush the curve", or suppression, strategy. We exhibit the terms of the trade-off between the loss of lives and the loss of GDP for the whole set of mitigation strategies, from laissez-faire to suppression. We compute the optimal mitigation policy as a function of the relative value the planner assigns to life over the economy, and the optimal prevention strategy, depending on the former and on the risk parameters.

The idea of assessing the economic impact of disasters that arise from "environmental" causes is the source of a well-stocked literature, both in the empirical works (see for instance [START_REF] Noy | The macroeconomic consequences of disasters[END_REF], and the contained references) and from the theoretical point of view (see for instance [START_REF] Akao | A theory of disasters and long-run growth[END_REF]Sakamoto, 2018 or Bakkensen and[START_REF] Bakkensen | Do disasters affect growth? A macro model-based perspective on the empirical debate[END_REF]. The general idea of these contributions is to understand the impact of environmental disasters on growth and development. Particularly inspiring for our research are the papers by Bretschger andVinogradova (2016, 2019) and [START_REF] Douenne | Disaster risks, disaster strikes, and economic growth: The role of preferences[END_REF]. The latter in particular studies the possibility of dealing with disaster of endogenous probability 5 and then the idea of disaster prevention. A related contribution is the recent paper 5

The idea of preserving biodiversity to reduce the probability of future negative outcomes is also used in another bunch of works in the literature, see for instance [START_REF] Baumgärtner | The insurance value of biodiversity in the provision of ecosystem services[END_REF], Baumgärtner and by [START_REF] Brock | The economy, climate change and infectious diseases: links and policy implications[END_REF] where the authors model the possible appearance of a shock linked to the arrival of an epidemic whose probability depends on the capital accumulated in the economy and the temperature anomaly caused by the use of nonrenewable resources. In the present paper the focus is different since, for the first time we link biodiversity conservation, the risk of zoonotic pandemics, population and economic dynamics and mitigation policies.

The paper is organized as follows. Section 2 introduces the model's assumptions and formulation. In Section 3 we describe the explicit solution of the model. Section 4 contains numerical simulations while Section 5 concludes. All the proofs are reported in Appendices A and B.

The model

We consider an economy where the planner makes decisions about how to manage biodiversity and how to deal with a pandemic if it happens.

2.1. Biodiversity and the economy. The planner decides at the beginning of the planning horizon how to allocate the land (whose total size is normalized to 1). A part f is devoted to biodiversity conservation while a part 1 -f is used for various human economic activities, such as agriculture, industry, human settlement and infrastructure.

Assimilating biodiversity and land devoid of human activity f can be justified both empirically and theoretically. Empirically, [START_REF] Newbold | A global model of the response of tropical and sub-tropical forest biodiversity to anthropogenic pressures[END_REF] show that habitat loss and degradation are the main causes of reductions in species richness and abundance (see also [START_REF] Wwf | Living planet: Report 2016: Risk and resilience in a new era[END_REF], IPBES, 2019). It has also been proved [START_REF] Brummitt | Green Plants in the Red: A Baseline Global Assessment for the IUCN Sampled Red List Index for Plants[END_REF] that habitat conversion and harvesting are the main threats to plant species in tropical forests. Land use change itself is driven by agricultural expansion, the expansion of urban area (which has doubled since 1992) and the expansion of infrastructure which has come mostly at the expense of forests. For this reason it is reasonable to consider the area of land that is not used for human activities f as a (proxy of a) measure of biodiversity. This idea was largely exploited in theoretical contributions. Already a century ago [START_REF] Arrhenius | Species and Area[END_REF] and [START_REF] Gleason | On the relation between species and area[END_REF] introduced the idea of species-area curves describing the link between the size of a habitat and the number of species living in the same habitat. This approach was refined over the decades (see for instance the presentation in the book of [START_REF] Rosenzweig | Species diversity in space and time[END_REF]. In line with [START_REF] Brook | Catastrophic extinctions follow deforestation in Singapore[END_REF] one can quantify the impact on habitat loss on biodiversity using the power relation s = ca z , where s is the ratio of original to current species, a is the ratio of original to Quaas (2010), [START_REF] Baumgärtner | The economic insurance value of ecosystem resilience[END_REF], [START_REF] Augeraud-Véron | The value of biodiversity as an insurance device[END_REF], 2020) which highlight the insurance value of biodiversity.

current habitat and c and z are constants. The effort to correctly calibrate this relation continues in recent studies: Storch et al. (2012) estimate a value of z = 0.21, while, for instance for Southeast Asian, [START_REF] Sodhi | A meta-analysis of the impact of anthropogenic forest disturbance on Southeast Asia's biotas[END_REF] and [START_REF] Wilcove | Navjot's nightmare revisited: logging, agriculture, and biodiversity in Southeast Asia[END_REF] find, respectively, 0.18 and 0.35 (see also [START_REF] Symes | Combined impacts of deforestation and wildlife trade on tropical biodiversity are severely underestimated[END_REF].

The choice of f influences the economy in two ways. Firstly in a direct way: the land devoted to human activities is used as an input in the production process. Indeed we suppose that the production is described by an aggregate production function of the form (1)

Y (t) = F (1 -f (t), A(t)N (t))
where N (t) is the size of the population and A(t) is labor productivity at time t. We abstract in particular from the use of capital (but the model can be extended in various ways to a more general production structure). We also abstract from describing the age structure of the population. Of course this can be a very important aspect, especially in situations where the consequences of the virus in terms of severity of the disease and morbidity change consistently depending on the age of infected people, as it is the case for the COVID-19 (see for instance Salje et al., 2020). 6 The homogeneous population modeling we use cannot account for this fact. It may nevertheless be justified in a setting like ours where different viruses cause epidemics of different nature, with a different impact on age groups. Consistently with the choice of not modeling factor accumulation and then investment, we suppose that all production is consumed:

(2)

C(t) = Y (t).
The second way the choice of f influences the economy is through its effects on zoonoses' outbreaks. We suppose that f influences the probability h(t) that a zoonose appears and becomes a pandemic: the bigger f , the more biodiversity and the smaller the probability of epidemic outbreaks. In this sense, biodiversity conservation is a risk-reducing policy.

2.2.

Population and productivity dynamics. The evolution of the population over time is modeled using a standard SIR model, as shown below. Absent epidemic outbreaks, population increases at a constant rate. When an epidemic outbreak hits, because of the different orders of magnitude of the parameters involved, the SIR epidemiological dynamics converges very fast in the neighborhood of the disease-free equilibrium manifold, and then is approximated by the exponential growth until either a 6 Macro-dynamic models which incorporate an age-structured epidemiological dynamics are for instance proposed by [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF], [START_REF] Fabbri | Verification Results for Age-Structured Models of Economic-Epidemics Dynamics[END_REF] and [START_REF] Favero | Restarting the economy while saving lives under Covid-19[END_REF].

re-emergence of the disease takes place, or a new emerging infectious disease appears. We only consider the second possibility, assuming that vaccines will prevent the disease of becoming endemic in the population.

Epidemic outbreaks are discrete events happening at time (τ j ) j∈N . We assume that the number M j of epidemic events satisfies a standard Poisson Process with instantaneous arrival rate h(f ). For any time t between two events [τ j , τ j+1 [, the dynamics is given by a SIR epidemic model in which population is decomposed into susceptibles S, infected I and removed R.

Population grows at rate n = ν -µ where ν is the birth rate and µ the natural mortality rate. Total population is given at date t by

N (t ) = S (t ) + I (t ) + R (t ) .
The transmission rate of the disease is denoted β. The recovery rate is r. The diseaseinduced mortality rate is δ. The epidemiological dynamics then reads:

(3)

           dN (t ) dt = nN (t ) -δI (t ) dS(t ) dt = νN (t ) -βS (t ) I(t ) N (t ) -µS (t ) dI(t ) dt = βS (t ) I(t ) N (t ) -(δ + r + µ) I (t ) dR(t ) dt = rI (t ) -µR (t ) with initial conditions N (τ j ) = N τ - j , S (τ j ) = N (τ j ), I (τ j ) = I 0 and R (τ j ) = 0.
For a large variety of zoonoses, demographic parameters n, ν and µ are very small compared to epidemiological parameters β and r, and the virus-induced death rate δ is small compared to β and r. It is the case for example for COVID-19. 7 Taking into account these different orders of magnitude enables us to rewrite the dynamics at different time scales, in particular at fast time scale and at slow time scale. 9 In Lemma 2.1, we show, using the fast time scale, that the dynamics quickly converges to the disease-free equilibrium manifold (also called the slow manifold) where the number of infected I is very low (step 1). Moreover, the fast dynamics appears to be a regular perturbation of an unperturbated system with first integrals. This enables us to compute the final size of the population [START_REF] Brock | The economy, climate change and infectious diseases: links and policy implications[END_REF]Xepapadeas, 2020, Augeraud-Véron, 2020 for the COVID-19 epidemics). Step 3 of the proof explains this computation. Using slow time scale, the motion on the slow manifold is also described (step 2).

Lemma 2.1. Following an epidemic outbreak at date τ j , the dynamics of system (3) converges quickly to (N (τ

j + ) = kN (τ j ), S(τ j + ) = N (τ j + ), I(τ j + ) = 0, R(τ j + ) = 0),
where k ∈ (0, 1) depends on the transmission rate β, the duration of the disease 1/r 7 For France, n = 5.7.10 -7 per day and µ = 2.5.10 -5 per day in 2019, 8 β = 0.274, γ = 0.1667 (http://www.data.gouv.fr) and δ = 1.44.10 -3 [START_REF] Carcione | A Simulation of a COVID-19 Epidemic Based on a Deterministic SEIR Model[END_REF].

9

See section 4 in [START_REF] Augeraud-Véron | Seasonal dynamics in an SIR epidemic system[END_REF].

and the virus-induced mortality rate δ. Then the slow motion dN (t) dt = nN (t), with S(t) = N (t), I(t) = 0 and dR(t) dt = 0 takes place, until the next epidemic outbreak.

Proof. The proof is given in Appendix A.

Consequently, the population dynamics considered in the model is a growth at constant rate n between the epidemic outbreaks, interrupted by instantaneous drops of the population of size 1 -k at each date τ when an epidemic hits:

(4) Ñ (τ ) = kN (τ )
and ( 5)

dN (t) N (t) = ndt -(1 -k)dq(t)
where q is a Poisson process with E [dq] = h(f )dt, with h (f ) < 0 : as justified above, the probability of a pandemic outbreak is all the lower since biodiversity is important.

Likewise, we assume that labor productivity grows at the exogenous and constant rate g > 0 between the epidemics outbreaks, and decreases instantaneously each time an epidemic hits the economy:

(6) Ã(τ ) = κA(τ )
with 0 < κ < 1. Thus the dynamics of productivity is described by the following stochastic differential equation:

(7) dA(t) A(t) = gdt -(1 -κ)dq(t).
2.3. Pandemics and policy response. The transmission rate of the disease β depends on the intensity of contacts among individuals within the population: the more intense social interactions are, the larger the transmission rate. Let e ∈ [0, ē] represent the intensity of social interactions. In normal times, this intensity is ē. The planner, by imposing a more or less severe and lengthy lockdown, is able to make the intensity of interactions decrease, eventually down to 0, which corresponds to a total lockdown. Therefore, the transmission rate β is a decreasing function of e. As a consequence, the instantaneous death rate 1 -k and the share of the population still alive after an epidemic outbreak k, depend on the policy response of the planner e as well, as stated in Lemma 2.2.

Lemma 2.2. The share of the population still alive after an epidemic outbreak, k, depends on the severity of the lockdown e. It is a function k(e) ∈ (0, 1), with k (e) < 0: the more severe the lockdown is (the smaller e) the fewer deaths in the population (the larger k(e)).

Proof. The proof is given in Appendix A.

We suppose more precisely that k(ē) = k ∈ (0, 1), k(0) = k ∈ (k, 1), and k ≤ 0: the marginal effectiveness of the mitigation policy is weakly decreasing (see Figure 1).

The drawbacks of the policy response to the pandemics is that it reduces labor productivity. "Non-essential" economic activities, that is activities outside the health, food and energy sectors, are stopped when teleworking is not possible. 10 Whenever it is possible, the potential issue is that it may transform from "working at home" to "shirking at home". [START_REF] Bloom | Does Working from Home Work? Evidence from a Chinese Experiment[END_REF] show, using an experiment performed on the 16,000 employees of a Chinese travel agency, that it is not the case: they report an increase in productivity of 13%. However, for teleworking to improve or be neutral towards productivity good working conditions at home are mandatory, like no small children to take care of simultaneously, a separate working environment, fast and reliable broadband services etc. [START_REF] Bloom | How working from home works out[END_REF].

Remark that, as the papers by [START_REF] Ruhm | Are recessions good for your health?[END_REF], [START_REF] Stevens | The best of times, the worst of times: understanding pro-cyclical mortality[END_REF] and [START_REF] Adda | Economic activity and the spread of viral diseases: Evidence from high frequency data[END_REF] show, the link between economic activity and diseases is sometimes (depending on the disease) pro-cyclical. This interesting and somehow surprising pattern does not undermine our model structure where the reduction in economic activity is not exogenous but is driven by a mitigation policy aimed at reducing the death toll of the pandemics, and so where it is natural that a drop in productivity corresponds to a reduction of mortality.

According to these observations we suppose that parameter κ in equation ( 6) is an increasing function of e: the more severe the lockdown the larger the productivity loss. We suppose more precisely that κ (e) > 0, κ(0) = κ ∈ (0, 1), κ(ē) = κ ∈ (κ, 1) and κ (e) ≤ 0 (see Figure 1).

Of course, the damages that the lockdown causes to the economy will not only come from a lower labor productivity. In particular production and supply chains will be disrupted directly, firms will get bankrupt in sectors heavily affected by the decrease of demand, etc. We make all these effects transit through a shock on labor productivity as a shortcut dictated by the fact that labor and land are the only inputs in our stylized model.

Finally, the only consequences of the lockdown policies considered here are mortality reduction on the one hand, and activity reduction through productivity losses on the 10 For instance [START_REF] Dingel | How Many Jobs Can be Done at Home?[END_REF] find that only 37% of jobs in the United States can be performed entirely at home. [START_REF] Bloom | How working from home works out[END_REF] finds that people who are able to telework are "mostly managers, professionals, and financial workers who can easily carry out their jobs on their computers by videoconference, phone, and email." other hand. However, a positive effect of the lockdown on biodiversity itself, due to reduced human pressures on wild species, has been widely observed in the case of COVID-19. The reduction in human mobility during the lockdown, relatively similar and simultaneous in a majority of countries, is unparalleled in recent history. The pressures on wildlife, and the environment in general, have been drastically reduced almost everywhere around the globe. Air quality has improved, CO 2 emissions have dropped, and there is ample anecdotal evidence of wild species thriving in these new conditions. However, this alleviation of the pressures is a short-term phenomenon, and during the recovery of the economy conditions are probably going to return to "normal". Therefore we choose not to make f depend on e.

2.4. Social welfare. Choosing a social utility functional is delicate since population ethics is a difficult matter (see for instance [START_REF] Arrhenius | The Repugnant Conclusion[END_REF]. We consider here a class of social welfare functions parameterized by λ ∈ [0, 1] by supposing that the planner's instantaneous utility at time t is of the form

N (t) λ u C(t) N (t)
where u C(t)

N (t)
is the utility derived from per-capita consumption. The value of λ measures the degree of altruism towards individuals of future cohorts (see for instance e.g. [START_REF] Boucekkine | Egalitarianism under population change: age structure does matter[END_REF] and it can be justified, as done by [START_REF] Barro | Fertility Choice in a Model of Economic Growth[END_REF] in a model of endogenous fertility choice by the "impure altruism" of parents. The two polar cases λ = 0 and λ = 1 correspond respectively to standard average utilitarianism and total utilitarianism.

The planner maximizes a non-separable intertemporal utility à la Epstein-Zin-Weil, supposing that agents' preferences are characterized by a constant relative risk aversion θ > 0 (and θ = 1), an intertemporal elasticity of substitution 1/φ > 0 (φ represents aversion to fluctuations) and a discount rate ρ > 0.

We can dig a little deeper in the utility functional to understand which choices of parameters are meaningful. If we look at the analogous of the informal representation of preferences given for instance by [START_REF] Svensson | Portfolio choice with non-expected utility in continuous time[END_REF] (see also [START_REF] Augeraud-Véron | The value of biodiversity as an insurance device[END_REF] in our case, we have, 11 for given controls e(t) and f (t), ( 8)

U (t) = 1 1 -φ N (t) λ C(t) N (t) 1-φ + e -ρdt E t U (t + dt) 1-θ 1-φ 1-φ 1-θ .
With this CRRA specification of the utility derived from per capita consumption we have to impose φ ∈ (0, 1) to make sure that the planner values positively an increase of the population.

The optimal policy

In order to explicitly solve the problem we specify the described setting by considering a linear probability of pandemics and a linear production function.

The instantaneous arrival rate of the Poisson process is then:

h (f ) = ε(1 -f ), ε ∈ (0, 1].
We acknowledge that this specification is not entirely satisfactory. Damages, in terms of occurrence of epidemic outbreaks, caused by the destruction of biodiversity would be better modeled as convex, or even better as featuring discontinuities and tipping points. We stick to this linear specification in order to obtain close-form solutions.

The production function reads:

Y (t) = (1 -f (t))A(t)N (t).
Then per-capita consumption is :

C(t) N (t) = (1 -f (t)) A(t)
and the instantaneous utility function becomes:

1 1 -φ N (t) λ C(t) N (t) 1-φ = 1 1 -φ N (t) λ (1 -f (t))A(t)) 1-φ .
11 As clarified by [START_REF] Epaulard | Optimally eating a stochastic cake: a recursive utility approach[END_REF] (see in particular equation (2.1) and Footnote 2)

this expression is equivalent to the one originally proposed by Svensson via a transformation à la Duffie and Epstein (1992).

Observe that, abstracting from dynamics effects, at the level of the instantaneous utility function, per-capita consumption does not decreases when N increases so no "quality/quantity" trade-off takes place. This is due to the linear specification of the production function.

All in all the HJB equation of the problem reads:

(9) ρ 1 -θ 1 -φ V (A, N ) = max f,e N λ [(1 -f )A] 1-φ 1 -φ 1 ((1 -θ)V (A, N )) 1-φ 1-θ -1 + V A gA + V N nN + ε(1 -f ) (V (κ(e)A, k(e)N ) -V (A, N ))
Theorem 3.1. Suppose that the function of the variable e ∈ [0, ē]

(10) k(e) λ 1-φ κ(e)
has a unique maximum point of e * ∈ [0, ē]. Suppose that the following interior condition

(11) ρ > (1 -φ)g + λn
is verified together with the following transversality condition:

(12) (1 -θ + φ)ρ > 1 -θ 1 -φ ((1 -φ)g + λn) .
Then optimal policy (f * (•), e * (•)) is deterministic and constant and it is given by e * (•) ≡ e * and f * (•) ≡ f * defined by: 12

(13)

f * := 1 -min   1, ρ -(1 -φ)g -λn φε 1 -θ 1 -k(e * ) λ 1-φ κ(e * ) 1-θ    .
The corresponding social welfare is given by

V (A, N ) = X N λ 1-φ A 1-θ 1 -θ with (14) X =     φ φ 1-φ (1 -θ) (ρ -(1 -φ)g -λn) φ 1-φ ε 1 -k(e) λ 1-φ κ(e) 1-θ     1-θ Proof. See Appendix B.
12 Observe that, since κ(e), k(e) ∈ (0, 1) then the term

(1-θ) 1 -k(e * ) λ 1-φ κ(e * ) 1-θ -1
is positive for any choice of θ > 0.

Theorem 3.1 first allows us to learn about the optimal mitigation policy. Denote by z(e) the function of e defined in equation ( 10). With our assumptions on the functions k and κ, z(e) is a positive and concave function of e: z (e) ≤ 0.

If z (e) < 0 for all e ∈ [0, ē], the maximum of z is attained for e = 0. The optimal policy is total lockdown: e * = 0. The condition z (e) < 0 is equivalent to λ

1-φ k (0) k(0) + κ (0) κ(0) < 0, that is -κ (0)/κ(0) k (0)/k(0) < λ 1-φ . Denote τ := -κ (0)/κ(0) k (0)/k(0)
. The case under study is then relevant for λ 1-φ > τ , that is for large values of the society's relative preference for life. Symmetrically, if z (e) > 0 for all e ∈ [0, ē], the maximum of z is attained for e = ē and the optimal policy is no lockdown: e * = ē. This case is relevant for λ 1-φ < τ , where τ := -κ (ē)/κ(ē) k (ē)/k(ē) , that is for small values of the relative preference for life. Finally, the optimal mitigation policy is interior for values of the relative preference for life between τ and τ . e * is then given by the first order condition for the maximization of ( 10 If the pandemic occurs the planner will react instantaneously by no/partial/total lockdown, according to her relative preference for life over the economy λ 1-φ , and the shape of the k and κ loss functions. On the left side of equation ( 15) we can recognize the opposite of the ratio between the elasticities of κ and k with respect to e. This expression can also be rewritten as -(1 -φ)κ (e)/κ(e) = λk (e)/k(e). Unsurprisingly in this expression the contribution of the elasticity of the negative effect on the population is weighted with λ. Indeed, as already mentioned, the per-capita consumption is not affected by the shock and the negative effect of the population loss only comes through the term N λ appearing in the utility function. This underlines once more the role of λ as a measure of the value of a life in the planner functional. Similarly the shock on productivity is weighted by the exponent (1-φ) of the productivity in the instantaneous utility. Risk aversion plays no role here. This is not particularly surprising because the choice of e has some impact on the system only after (and if) the shock occurs. Conversely the value of θ clearly appears in the choice of f which is the "disaster prevention choice" of the planner. Due to the linearity of the model, the growth rates of the productivities (i.e. g and n) do not enter in the choice of e.

Turning to the biodiversity conservation policy, we obtain several results by inspection of equation ( 13).

First, the share of land optimally devoted to biodiversity conservation is a decreasing function of the discount rate: the more the planner is interested in future outcomes, the more she wants to reduce the risk of pandemic outbreaks and the more biodiversity is needed. Conversely the higher the inherent capacity of the system to regenerate after a shock (measured by the deterministic growth rates of the productivity and the population g and n) the smaller the optimal biodiversity conservation.

Then, the role of the parameter ε describing the relationship between biodiversity and the probability of a pandemic is straightforward; a larger ε leads to more biodiversity conservation, which is conform to intuition. Indeed, the larger ε the more powerful is biodiversity conservation in terms of risk reduction.

As for the preference parameters the results are not obvious and are collected in the following proposition.

Proposition 3.2. Suppose that the hypotheses of Theorem 3.1 are verified and that the optimal value of f * is interior. Then f * is an increasing function of θ and λ. Moreover it is an increasing function of φ when the discount rate is high enough: ρ ≥ g + λn.

Proof. See Appendix B.

Even if it is not obvious at a first look the impact of risk aversion is conform to intuition: f * is an increasing function of risk aversion, meaning that the more risk averse the planner is, the more she wants to reduce the risk of future pandemics.

Similarly Proposition 3.2 tells us that f * is an increasing function of λ.

We can dig a bit further here. In the limit case of average utilitarianism (λ = 0), the optimal mitigation policy is no lockdown (e * = 0), since λ 1-φ = 0 < τ . But is it necessarily the case that the optimal biodiversity conservation is nil? We have:

(16) f * | λ=0 = 1 -min 1, ρ -(1 -φ)g φ 1 -θ 1 -κ 1-θ
If the productivity loss absent any social distancing is very small, κ is close to 1 and f * = 0: there is no biodiversity conservation. Conversely, if the pandemic causes a significant productivity loss even without lockdown (and not only loss of lives), conserving some biodiversity may be optimal to reduce the risk of pandemic outbreaks.

Moreover, we can infer from the previous results the nature of the links between optimal prevention and optimal mitigation. At the beginning of the planning horizon, the planner chooses the optimal mitigation policy she will put in place each times a pandemic hits. This policy is all the more severe since she values life a lot. Therefore, the loss she incurs each time a pandemic hits, 1 -k(e * ) λ 1-φ κ(e * ) is all the larger (see the proof of Proposition 3.2). To avoid paying this large cost too often, the planner will choose to conserve a lot of biodiversity.

Finally Proposition 3.2 emphasizes that f * is an ambiguous function of the aversion to fluctuations φ but a sufficient condition for f * being an increasing function of aversion to fluctuations is that the discount rate is high enough: ρ ≥ g + λn. In the case of average utilitarianism (λ = 0), this condition reads ρ > g and is necessary and sufficient. Indeed we have, using equation ( 16) when the solution is interior:

∂f * ∂φ | λ=0 = 1 -θ 1 -κ 1-θ ρ -g φ 2 .
where the term 1-θ 1-κ 1-θ is positive for any choice of θ since κ ∈ (0, 1).

Corollary 3.3. Let hypotheses of Theorem 3.1 be satisfied. Then the optimal evolution of A(t) and N (t) are A(t) = A 0 e gt κ(e * ) q(t)-q(0)

N (t) = N 0 e nt k(e * ) q(t)-q(0)
where the increment q(t) -q(0), that is the number of pandemic outbreaks since date 0, is Poisson-distributed with mean ε(1 -f * ). In particular

E(A(t)) = A 0 e (g-ε(1-f * )(1-κ(e * )))t , V ar(A(t)) = (E(A(t))) 2 e (g-ε(1-f * )(1-κ(e * )))t -1 E(N (t)) = N 0 e (n-ε(1-f * )(1-k(e * )))t , V ar(N (t)) = (E(N (t))) 2 e (n-ε(1-f * )(1-k(e * )))t -1
The expressions of previous corollary are particularly transparent: the evolution of A(t) and N (t) only depends on their pandemic-free (exponential) dynamics and on the size and the number of shocks (which, in the model, always have the effect of reducing the quantities by the same factor). The growth rates of the expected productivity and population are equal to the pandemic-free growth rates (g and n respectively) adjusted for the effects of possible pandemics given by the size of the loss weighted by the probability of the pandemics. A large weight of life in the social welfare function (large λ) causes a severe optimal mitigation policy (small e * ) and thus a large drop of productivity and a small death toll, together with an important risk-reducing effort of biodiversity conservation (large f * ). Therefore, the growth rate of the expected population is unambiguously an increasing function of λ, whereas the effect of λ on the growth rate of expected productivity is ambiguous.

Numerical illustration

4.1. Specifications. We use the following specifications for the k and κ functions:

k(e) = k -k -k e ē a κ(e) = κ + (κ -κ) e ē b
When a ≥ 1 and 0 < b ≤ 1 these specifications satisfy the assumptions made above. 2020) calibrates a SIR model on the COVID-19 pandemic data for France and performs cost-benefit analysis exercises. He estimates the death toll and the GDP loss in the no-policy case, and for two strategies in particular: the "suppression", or "crush the curve" strategy, consisting in confining 90% of the population for 4 months to eradicate the virus, and the "flatten-the-curve" strategy, consisting in confining 30% of the population for 5 months. We use these data for our benchmark calibration. Needless to say, the parameters used by Gollier to calibrate the SIR model and the results he obtains are highly uncertain. We report in Table 1 Gollier's results and also the results of two other papers, obtained for different countries and with different methods, to assess whether they give similar or very different information.

Greenstone and Nigam (2020) are only interested in the death toll of the pandemic. Using the famous [START_REF] Ferguson | Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand[END_REF] simulation model of COVID-19's spread and mortality impacts in the United States, they estimate the death toll in the laissez-faire situation and in the case of a moderate social distancing policy, taking into account, as Gollier does, not only direct deaths but also the deaths due to the overwhelming of hospital intensive care units. [START_REF] Thunström | The benefits and costs of using social distancing to flatten the curve for COVID-19[END_REF] also examine the impacts of social distancing in the US, but on both the death and the GDP sides. They use epidemiological and economic forecasting to perform a cost-benefit analysis of controlling the COVID-19 outbreak. We assume here that Gollier's "flatten the curve" scenario, Greenstone and Nigam "moderate social distancing" scenario and Thunström et al. "control" k(e) κ(e) laissez-faire k = 1 -0.0054

κ = 1 -0.0373 mitigation k -k -k e a m = 1 -0.00363 κ + (κ -κ) e b m = 1 -0.0574 suppression k = 1 -0.00029 κ = 1 -0.1353
Table 2. Calibration of the loss functions scenario are roughly equivalent in terms of severity and length of the lockdown. Table 1 shows that Gollier and Thunström et al. give similar estimates of the death toll in the laissez-faire scenario (no policy) and the mitigation (flatten the curve) scenario whereas Greenstone and Nigam are more pessimistic. For the economic cost of the pandemic in terms of GDP loss, the results from Thunström et al. we report in the table correspond to immediate losses (the year of the outbreak). The authors also compute the present value of GDP losses on a 30-year horizon, that we do not use for comparability with Gollier's estimates. Gollier is more pessimistic in the laissez-faire scenario, less so in the mitigation scenario. But again theses estimates are very uncertain.

4.3. Calibration. We normalize ē to 1. We use Gollier's data, as reported in Table 1, to calibrate the parameters of the k and κ loss functions. The laissez-faire, mitigation and suppression scenarios correspond respectively to e = ē = 1, e = e m unknown and e = 0. We have to calibrate parameters k, k, κ, κ, e m , a and b. The relationships reported in Table 2 allow us to obtain 6 out of these 7 parameters. We choose to calibrate a/b, set arbitrarily a = 2 and deduce b. The results are in Table 3.

ē e m a b k k κ κ ρ g n φ 1
0.753 2 0.8 0.9946 0.99971 0.8647 0.9627 0.02 0.02 0.005 0.95 Table 3. Results of the calibration of the loss functions, and other parameters We obtain that the mitigation strategy corresponds to a moderate lockdown: social interactions are reduced by around one quarter (e m = 0.753).

The other parameters of the model, namely the discount rate ρ, the deterministic growth rates of population and productivity n and g, and the aversion to fluctuations φ are in the range of the parameters found in the literature (Table 3). 4.4. Results. Figure 2 shows the terms of the raw trade-off between the loss of lives and the loss of GDP, for a mitigation strategy between 0 (suppression), on the left end 2020) obtain in a model with several age classes. In their case, if, from the laissez-faire situation, the planner decides to confine the older age classes, most at risk, then she can at the same time save lives and mitigate the GDP loss. For more severe mitigation policies, all age classes are confined and the trade-off becomes similar to ours. -ε) frontier for λ = 0.9 φ = 0.95, the lower threshold corresponds to λ = 0.52. In the case of average utilitarianism (λ = 0) and for all λ < 0.52 it is not optimal to engage in mitigation. In the case of total utilitarianism (λ = 1) we obtain e * = 0.44: optimal mitigation is far more severe that in Gollier (2020)'s mitigation strategy for which e * = e m = 0.753, that is far more severe than the the "flatten-the-curve" strategy, consisting in confining 30% of the population for 5 months.

The very recent literature on the cost-benefit analysis of COVID-19 mitigation policies commonly uses the Value of a Statistical Life (VSL) to monetize the death toll and compare the benefits of the policy in terms of avoided deaths to its costs in terms of foregone GDP. We do not need to do that here. In our model, the relative value of life is V N N V A A , the value of the population over the value of productivity, population and productivity being both valued at the marginal increase of welfare their increase causes. With our specifications the relative value of life is constant and equal to λ 1-φ . It only depends on the characteristics of the social welfare function, the degree of partial altruism λ and the aversion to fluctuations φ.

The optimal biodiversity conservation f * , given by equation ( 13), depends on all the parameters identified above but also on the risk parameters, the risk aversion θ and the probability that the pandemic hits absent biodiversity ε. This last parameter is particularly difficult to calibrate. Instead of engaging in the exercise, we choose to determine the couples (θ, ε) constituting the frontier between no biodiversity conservation and biodiversity conservation or, to put things differently, no prevention and prevention of the pandemics. This frontier is represented on Figure 4 for λ = 0.9. We see that with our calibration, when risk aversion is around 2 biodiversity conservation becomes optimal for very high values of the probability of pandemics (around 12%), whereas when risk aversion is around 20 it becomes optimal for values of the probability of pandemics around 4%. Again, these figures are only illustrative. To compute the optimal prevention policy as a function of λ we choose θ = 10 and ε = 0.1. Figure 5 shows that there is no biodiversity conservation until λ = 0.6, and that in the case of total utilitarianism 35% of the land is devoted to biodiversity conservation for the prevention of pandemics.

Conclusion

Following the outbreak of the COVID-19 pandemic a long series of contributions in theoretical and applied economics have been published.

In this work we dig a little not only into the policies to be implemented to mitigate the effects of the pandemic when it occurs, but also (and above all) into one of the possible vectors to prevent it. We propose, for the first time as far as we know, a theoretical dynamic model that looks, in the case of a stylized economy, at the economic importance of reducing the likelihood of zoonotic outbreaks through the conservation of biodiversity.

We consider a family of instantaneous utility functions admitting per capita consumption and the size of the population as arguments, to be able to derive from preferences the choice society has to make between economic activity and lives' preservation when an epidemic outbreak happens.

Despite a certain technical complexity (due to the presence of jump processes and of an Epstein-Zin type social welfare functional) the model is completely solved and a discussion about the impact of various elements at stake (situation of the natural environment, productivity, preferences) is developed.

Emphasizing a role of "forward looking" instrument, we prove that biodiversity conservation is more important when the social welfare functional is characterized by a lower discount rate and a stronger degree of altruism towards individuals of future cohorts. Not surprisingly, given the risk-reducing (prevention) effect of biodiversity in the model, biodiversity conservation is also more relevant when the risk aversion or the risk of pandemics are higher.

After calibrating the model using the data from Gollier (2020) we exhibit the terms of the trade-off between the loss of lives and the loss of GDP for the whole set of mitigation strategies, from laissez-faire to suppression.

In this first attempt to look, in a macrodynamic model, at the economic consequences of the effects of the destruction of biodiversity on the probability of developing zoonoses, we considered the problem in a simplified setting that allow us in particular to obtain an explicit solution to the optimization problem. We believe that the essential features of the problem and the essential trade-offs are maintained in this formulation. There remain, however, very relevant aspects of the problem that cannot be described in our set-up. They could be interesting extension of our research.

First, we have already mentioned the absence of age structure of the population in the model, which is certainly an element of interest in the case of zoonoses that, as in the case of COVID-19, have a very different impact on different age groups and then on the best policies to be implemented (see for instance [START_REF] Acemoglu | A multi-risk SIR model with optimally targeted lockdown[END_REF].

Secondly, capital accumulation is absent in our model. Taking it into account would make the effects of containment policies persistent, as the decrease of labor productivity, causing a decrease of GDP, would lead to less capital accumulation, which would have long-lasting consequences (see [START_REF] Brock | The economy, climate change and infectious diseases: links and policy implications[END_REF]. Moreover, the epidemics could cause capital destruction, if activities disappear permanently because of a permanent change of the demand pattern.

Then, our model describes the world economy, and is unable to take into account globalization and the role of trade and travels in the spread of epidemics and the fact that they become pandemics [START_REF] Tatem | Global transport networks and infectious disease spread[END_REF].

Another element of interest that has been particularly highlighted during the COVID-19 outbreak is the differential effect of containment measures (and in particular lockdown) on different sectors of the economy, with different impact on productivity of different sectors and reallocation effects (see for example [START_REF] Krueger | Macroeconomic dynamics and reallocation in an epidemic[END_REF]. The food and agriculture sector could take the crisis as an opportunity to accelerate its transformation towards more resilience, which could at the same time improve its productivity and decrease its pressure on the environment. The crises produces strong incentives for more robots and digital technologies in industrial sectors, which has the potential to improve productivity. Besides, innovation may be boosted, through a creative destruction process. However, the long term perspectives are still unclear. These sectoral effects and their consequences on long term growth and innovation cannot be taken into account in a one-sector set-up as the one we consider here.

Finally, another important shortcoming of our analysis is that distributive issues are absent. The COVID-19 crisis has affected very differently people according to their age, but also to their income level. The differential effect across income categories is the addition of a direct effect (richer people have better access to the health system, live in larger houses and a less polluted environment, and are better protected against unemployment and the loss of their income) and an indirect effect through public policies (the lockdown affects more the jobs of poorer people), the poorest being more affected. 13 The stylized framework we have developed in this work would enable us to extend our research in several directions. First, introducing capital accumulation and considering more general production functions is clearly on he agenda. Second, we intend to consider a game-theoretical extension of the model to study the cost of non-coordination of policies at the world level. Finally, we would like to consider uncertainty, disentangling aversion to ambiguity, risk aversion and aversion to fluctuations. Indeed, the COVID-19 crisis has made clear how little we know on the biological and epidemiological sides, and also on the side of the appropriate mitigation policies.

Replacing 1 -f in the HJB equation (9) yields:

(27) ρ 1 -θ 1 -φ V = max e N λ A 1-φ 1 -φ N λ A 1-φ ε(V -Ṽ )((1 -θ)V ) 1-φ 1-θ -1 1-φ φ 1 ((1 -θ)V ) 1-φ 1-θ -1 + V A gA + V N nN + N λ A 1-φ ε(V -Ṽ )((1 -θ)V ) 1-φ 1-θ -1 1/φ ε( Ṽ -V ) = max e φ 1 -φ N λ A 1-φ 1 φ ε(V -Ṽ ) -1-φ φ ((1 -θ)V ) 1-φ 1-θ -1 -1 φ + V A gA + V N nN
We try to find a solution of the form:

V (A, N ) = X N α A 1-θ 1 -θ .
for some real positive parameters α and X. In this case

Ṽ = k(e) α κ(e) 1-θ V V A = (1 -θ) V A V N = α V N
Using these expressions in the HJB equation above we get:

ρ 1 -θ 1 -φ V = max e φ 1 -φ N λ A 1-φ 1 φ ε(1 -k(e) α κ(e) 1-θ )V -1-φ φ ((1 -θ)V ) 1-φ 1-θ -1 -1 φ + (1 -θ)gV + αnV i.e. ρ 1 1 -φ -g - α 1 -θ n = max e φ 1 -φ N λ A 1-φ 1 φ ε 1 -k(e) α κ(e) 1-θ 1 -θ -1-φ φ N α A 1-θ -1 φ 1-φ 1-θ X -1 φ 1-φ 1-θ
The maximum point for e is given by e * which maximizes (10). Choosing α = λ 1 -θ 1 -φ allows us to obtain:

1 1 -φ (ρ -(1 -φ)g -λn) = φ 1 -φ ε 1 -k(e * ) α κ(e * ) 1-θ 1 -θ -1-φ φ X -1 φ 1-φ 1-θ
i.e. finally (recall that assumption (11) implies ρ -(1 -φ)g -λn > 0):

X =       φ ε(1-k(e * ) α κ(e * ) 1-θ ) 1-θ -1-φ φ ρ -(1 -φ)g -λn       φ 1-θ 1-φ
Then (26) reads:

(28)

1 -f = 1 -θ ε (1 -k(e * ) α κ(e * ) 1-θ ) X 1-φ 1-θ 1/φ = (1 -θ)(ρ -(1 -φ)g -λn)
φε 1 -k(e * ) λ 1-φ κ(e * )

1-θ

Since condition ( 11) is verified this value is in (0, 1).

We finally compute the transversality condition. We need to obtain that lim t→∞ e -ρt (A(t), N (t)) = 0.

The optimal evolution of A (t) and N (t) are A(t) = A 0 e gt κ(e * ) q(t)-q(0) N (t) = N 0 e nt k(e * ) q(t)-q(0)

Then:

V (A(t), N (t)) = X 1 -θ N λ 1-θ 1-φ 0 A 1-θ 0 e
(1-θ)g+λ 1-θ 1-φ n t κ(e * ) (1-θ)(q(t)-q(0) k(e * ) λ 1-θ 1-φ (q(t)-q(0)

and

E   (1 -θ) V (A(t), N (t)) XN λ 1-θ 1-φ 0 A 1-θ 0   = e
(1-θ)g+λ 1-θ 1-φ n-1-κ(e * ) 1-θ k(e * ) λ 1-θ 1-φ ε(1-f * ) t

i.e., using the definition of f * ,

E   (1 -θ) V (A(t), N (t)) XN λ 1-θ 1-φ 0 A 1-θ 0   = e
(1-θ)g+λ 1-θ 1-φ n-1-θ φ (ρ-(1-φ)g-λn) t

So we have e -ρt EV (A(t), N (t)) → 0 as far as

-ρ + (1 -θ)g + λ 1 -θ 1 -φ n - 1 -θ φ (ρ -(1 -φ)g -λn) < 0
which reduces to the condition asked in (12).

Proof of Proposition 3.2. In the case of an interior solution on f * , equation ( 13) reads: P is therefore maximum for θ = 1, and as P (1) = 1 we can conclude that indeed P (θ) ≤ 1. f * is an increasing function of θ.

f * = 1 - ρ -(1 -φ)g -λn εφ 1 -θ 1 -z(e *
Regarding the dependence of f * on λ, we have: The sign of ∂f * ∂φ is therefore ambiguous. A sufficient condition for it to be positive is ρ ≥ g + λn, that is a high enough discount rate.

Figure 1 .

 1 Figure 1. Instantaneous responses to the pandemics as a function of the mitigation policy

Figure 2 .

 2 Figure 2. Trade-off lives vs economy

Figure 3 .

 3 Figure 3. Optimal mitigation policy as a function of λ

Figure 5 .

 5 Figure 5. Optimal prevention policy as a function of λ, for ε = 0.05 and θ = 8

  ) 1-θ with z(e * ) = k(e * ) λ 1-φ κ(e * )and e * does not depend on θ. Then∂f * ∂θ = (ρ -(1 -φ)g -λn)[1 -(1 -(1 -θ) ln z(e * ))z(e * ) 1-θ ] εφ(1 -z(e * ) 1-θ ) 2 ∂f * ∂θ ≥ 0 ⇐⇒ 1-(1-(1-θ) ln z(e * ))z(e * ) 1-θ ≥ 0 ⇐⇒ (1-(1-θ) ln z(e * ))z(e * ) 1-θ ≤ 1. Let P (θ) = (1 -(1 -θ) ln z)z 1-θ . P (θ) = ln zz 1-θ + (1 -(1 -θ) ln z) ∂e (1-θ) ln z ∂θ = (1 -θ)(ln z) 2 z 1-θ .P (θ) 0 ⇐⇒ θ 1.

  ρ -(1 -φ)g -λn)(1 -θ)z(e * ) -θ 1 -z(e * ) optimality condition defining the mitigation policy (equation (15)) the term in parenthesis on the right-hand side member of this equation is nil. Then∂z(e * ) ∂λ = z(e * ) 1 1 -φ ln k(e * ) < 0,from which we conclude that ∂f * ∂λ > 0. Finally, regarding the dependence of f * on φ, we have: -(1 -φ)g -λn)(1 -θ)z(e * ) -θ 1 -z(e * ) * ) ∂φ = z(e * ) 1 (1 -φ) 2 ln k(e * ) < 0.

Table 1 .

 1 Data

			Gollier Greenstone Thurnström
				and Nigam	et al.
		mortality
	Laisser-faire (% population) 0.54	1.13	0.68
		economic cost
		(% GDP)	3.73		2
		mortality
	Mitigation	(% population) 0.363	0.52	0.29
		economic cost
		(% GDP)	5.74		6.2
		mortality
	Suppression (% population) 0.029
		economic cost
		(% GDP)	13.53

4.2. Data. Gollier (

Acknowledgements

The work of Giorgio Fabbri is supported by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02) and of the center of excellence LABEX MME-DII (ANR-11-LBX-0023-01)

https://www.centerforhealthsecurity.org/our-work/events

To study this dynamics, we need to take into account the orders of magnitude of the parameters. According to the data, the growth rate n, birth rate ν and mortality rate µ of the population are small compared to the virus-induced mortality rate δ, which is small compared to the other parameters of the epidemics β and r.

We thus rewrite n = εη n, ν = εη ν, µ = εη µ, and δ = η δ, where 0 < η << 1 and 0 < ε << 1. Now parameters n, ν, µ and δ are of the same order of magnitude. We also consider three time scales: a fast time scale t , an intermediate time scale σ and a slow time scale t, defined as t = εηt and t = ησ.

Step 1 of the proof Introducing the change of variable y = i/ε and skipping the time variable, system (17) can be written as:

= βs -η δ -r -εη ν + εη δy y

Setting ε = 0 enables to consider system (18) as a regular perturbation of the unperturbed system (19) given by:

The informations conveyed by system ( 19) are the following. At fast time, the system expressed in variables (N, s, y) is such that only the y dynamics matters. At this time scale, with this choice of variables, s and N are quite constant. Let s = s j the constant value of s. Manifold y = 0 (called in the following the slow manifold, because it would be the locus of the motion in slow time) is attractive if s j < r+δ β = 1 R 0 , where R 0 is the reproduction ratio of the disease. This slow manifold is also called the disease-free equilibrium. According to Tikhonov's theorem, a trajectory of system (18) satisfying s j < 1 R 0 , quickly jumps to the slow manifold y = 0, where the slow motion takes place.

Step 2 of the proof We now consider the slow motion of system (18) on the slow manifold y = 0. To do so, let us write system (18) using slow time t:

Setting ε = 0 and η = 0 enables to consider system (20) as a singular perturbation of the unperturbed system (21) given by:

On the slow manifold, the dynamics satisfies dN (t) dt = nN (t) . Thus the growth rate of the population is constant. It will remain constant until the slow manifold becomes unstable or a new epidemic outbreak occurs.

Step 3 of the proof It now remains to explain why there exists s j such that the slow manifold is attractive. To do this, let us come back to the original fast time scale system (17). Setting ε = 0 and η = 0 enables to consider this system as a regular perturbation of the unperturbed system (22) given by:

First integral of the system can be used to compute the final size of the epidemics, which is defined as the number s ∞ of per capita susceptibles at the end of the epidemic outbreak. s ∞ is solution of the following equation:

As r/β 1/R 0 since δ is very small compared to r and β, this equation can also be written on the more standard following form:

The share of the population 1 -s ∞ that has been infected is either recovered or dead. The per capita number of deaths is then δ r+δ (1 -s ∞ ). Thus the final size if the population after the fast epidemic outbreak is kN (t j ) , with

Moreover, as ds(t )/dt < 0, there exists an instant t " such that s (t") = s j < 1 R 0 . Thus according to step 2 the dynamics quickly jumps to the slow manifold, where the slow motion then takes place.

Proof of Lemma 2.2. In the case where the transmission rate is an increasing function of the intensity of social interactions, β(e), with β (e) > 0, the reproduction number R 0 = β r+δ is an increasing function of e as well, and the implicit function theorem, applied to equation ( 23) yields:

which is negative as soon as s ∞ < 1 R 0 , which is the case. Thus, according to equation ( 24),

Appendix B. Proofs of Theorem 3.1 and Proposition 3.2

Proof of Theorem 3.1. As already mentioned in the main text the HJB equation related to our maximization problem is

Denote Ṽ = V ( Ã, Ñ ). The optimality condition w.r.t. f yields:

(26)


Raouf Boucekkine

Giorgio Fabbri

Salvatore Federico

Fausto Gozzi

CONTROL THEORY IN INFINITE DIMENSION FOR THE OPTIMAL LOCATION OF ECONOMIC ACTIVITY: THE ROLE OF THE SOCIAL WELFARE FUNCTION

Keywords: 2010 Mathematics Subject Classification. 49L20, 35Q93, 47D06, 34G10 Spatiotemporal growth models, Benthamite vs Millian social welfare functions, imperfect altruism, optimal control of PDE's, Hamilton-Jacobi-Bellman equations in infinite dimension

come

Control theory in infinite dimension for the optimal location of economic activity: The role of social welfare function

Introduction

In this paper, we consider the problem of optimally control the linear abstract system in a Hilbert space H x (t) = Lx(t) -N c(t), where L, N are suitable operators, under the constraint that x(t), ϕ > 0, where •, • denotes the inner product in H and ϕ ∈ H is a given vector. The aim is to maximize, over the associates set of admissible controls, a functional in the form ∞ 0 e -ρt U(c(t))dt, where ρ > 0 and U is an utility function. To solve the problem, we use the dynamic programming technique in infinite dimension (see, e.g., Bensoussan et al. [START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF] and Li and Yong [START_REF] Li | Optimal Control Theory for Infinite Dimensional Systems[END_REF]. The method was implemented an economic modeling by Fabbri and Gozzi [START_REF] Fabbri | Solving optimal growth models with vintage capital: The dynamic programming approach[END_REF] in a different context, and then employed, in a similar context, in a previous work by Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]; precisely, we explicitly solve the HJB equation in infinite dimension associated to the optimization problem and then we prove a verification theorem and provide an optimal control in feedback form.

The problem is motivated by economic growth models with space dimension. There is indeed a growing interest in the economic literature in the role of space in decision-making. While the economic analysis has incorporated the spatial dimension for quite a long time (Von Thunen [START_REF] Von Thunen | Der Isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie[END_REF]), it is the rise of the so-called New Economic Geography (NEG) which did induce such a recent boom in this literature stream (see Krugman [START_REF] Krugman | What's new about the New Economic Geography?[END_REF] for an insightful review of the NEG, and Fujita and Thisse [START_REF] Fujita | Economics of agglomeration: Cities, industrial location, and regional growth[END_REF], for a master textbook in this area). An overwhelming part of the latter stream has been concerned with the identification and characterization of spatial externalities and the inherent agglomeration mechanisms. Concretely, researchers in the area target either first nature causes for agglomeration for given technology and demographic spatial distributions (Krugman [START_REF] Krugman | First nature, second nature, and the metropolitan location[END_REF]) or second nature causes through the identification of mechanisms (typically, economies of scale or spillovers) leading to agglomeration for example when labor is mobile (Krugman [START_REF] Krugman | Increasing returns and economic geography[END_REF]). An overwhelming majority of this paper is purely static: individuals do not save over time and therefore no capital accumulation is considered. Following an earlier contributions in mathematical geography by Isard and Liossatos [START_REF] Isard | Liossatos Spatial Dynamics and Optimal Space-time Development[END_REF], Brito [START_REF] Brito | The dynamics of growth and distribution in a spatially heterogeneous world[END_REF] is to our knowledge the first who attempted to insert space in otherwise standard neoclassical growth models, giving rise to a bunch of papers on optimal growth within spatiotemporal frames (in particular, Boucekkine et al. [START_REF] Boucekkine | Bridging the gap between growth theory and the new economic geography : The spatial Ramsey model[END_REF][START_REF] Boucekkine | Spatial dynamics and convergence: The spatial AK model[END_REF][START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], and Fabbri [START_REF] Fabbri | Geographical structure and convergence: A note on geometry in spatial growth models[END_REF]).

From the economic point of view, this paper builds on Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]. We focus on the problem faced by a central planner who has to choose the optimal distribution of economic activity (say, investment and production) over space (here the unit circle for simplicity). We assume that technology is homogeneous across space, which amounts to assuming that technological spillovers are quick enough for all the locations to use the same technology. In contrast, as it is most of the time the case, individuals are harder to move. We do not justify such an immobility but just assume it. In other words, the central planner can move capital but not people. Accordingly, our problem investigates a first nature cause of (potential) spatial externalities. Moreover, we generalize the social welfare function considered by Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] by introducing a form of imperfect altruism in the social preferences. While only the Benthamite social welfare function is considered in the latter (that is, the planner consider the sum of utilities of all the individuals present in the economy), we consider a continuum of social welfare functions ranging from the Benthamite to the Millian form (i.e only the average consumer is considered). To our knowledge, this is the first time this is done within a spatial setting. Clearly, this would allow to handle a set of new population ethics and political economy issues, for example concerning regional particularism. Earlier papers using this continuum of social welfare functions in non-spatial settings have been typically devoted to study the normative implications of 1984 Parfit's [START_REF] Parfit | Reasons and persons[END_REF] population ethics theory in different economic contexts (see Nerlove et al. [START_REF] Nerlove | Population size: individual choice and social optima[END_REF] 1985, Palivos and Yip [START_REF] Palivos | Optimal population size and endogenous growth[END_REF], and more recently, Boucekkine and Fabbri, [START_REF] Boucekkine | Assessing Parfit's Repugnant Conclusion within a canonical endogenous growth set-up[END_REF]).

Importantly enough, in the induced control problem, the capital spatiotemporal dynamics follows a parabolic PDE in the tradition opened by Isard and Liossatos [START_REF] Isard | Liossatos Spatial Dynamics and Optimal Space-time Development[END_REF], and can be lifted into our abstract setting. Hence, it is possible, at the end of the day, to individuate the optimal spatiotemporal control and dynamics.

The paper is organized as follows. Sections 2 to 5 give all the mathematical steps needed to solve explicitly the optimal control problem. Section 6 describes the economic application and apply the preceding mathematical theory. Section 7 displays some complementary numerical exercises. Section 8 concludes.

The optimal control problem

Let (D, D, µ) be a countably generated measure space and let us consider the separable Hilbert space H := L 2 (D, µ; R), with its usual norm | • | and scalar product •, • . We identify, by the usual Riesz identification, H with its topological dual. We shall identify the elements of H, which are equivalence classes of functions coinciding µ-a.e., with (one of) their representative functions. So, the pointwise relationship must be intended µ-a.e.. We consider the positive and strictly positive orthants of H, i.e. the sets

H + := {f ∈ H : f ≥ 0}, H ++ := {f ∈ H : f > 0}.
Finally, we denote H 0 + := H + \ {f ≡ 0}. In the following we set R + := [0, +∞).

Given this setting, we are concerned with the following optimal control problem in the space H. Let L : D(L) ⊆ H → H be a (possibly) unbounded linear operator and let N : H → H a linear bounded operator. Given x 0 ∈ H and a control function c ∈ L 1 loc (R + ; H), we consider the following abstract state equation in H:

(2.1) x (t) = Lx(t) -N c(t), x(0) = x 0 .
We introduce the following assumption Assumption 2.1. L is a closed and densely defined operator generating a C 0 -semigroup in H.

We use the notaton e tL for the semigroup generated by L and, according to Bensoussan et al. [START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF], we define the mild solution to (2.1) as the function (2.2) x(t) := e tL x 0 -t 0 e (t-s)L N c(s)ds.

To stress the dependence of x on x 0 , c, we write x x 0 ,c . (2.2) is also a weak solution to (2.1), i.e.

(2.3)

x(t), ψ = x 0 , ψ + t 0 x(s), L * ψ ds- t 0 N c(s), ψ ds, ∀ψ ∈ D(L * ), ∀t ≥ 0,
where

L * : D(L * ) ⊆ H → H denotes the adjoint of L. Assumption 2.2. u : D × R + → R + is such that u(θ,
•) is a utility function for each θ ∈ D -i.e. increasing and concave.

Next, given u : D × R + → R + as above u(θ, •), we consider the functional

U(z) := D u(θ, z(θ))µ(dθ), z ∈ H + ,
and the functional on

L 1 loc (R + ; H + ) (2.4) c → J (c) := ∞ 0 e -ρt U(c(t))dt,
where ρ > 0 is a given discount factor. Notice that both U and J inherit from u the concavity.

Remark 2.3. The choice of working only with nonnegative (or, equivalently, bounded from below) utility function is done here just for the sake of brevity, in order to avoid technical complications that would arise in treating the case of utility functions not bounded from below. The latter case might be considered and treated as well at the price of technical complications.

Assumptions 2.1-2.2 will be standing from now on. Let ϕ ∈ H ++ . The aim is to maximize U when c ranges over the convex set

A ϕ ++ (x 0 ) = {c ∈ L 1 loc (R + ; H + ) : x x 0 ,c (t) ∈ H ϕ ++ for a.e. t ≥ 0},
where

H ϕ ++ is the open set H ϕ ++ := {x ∈ H : x, ϕ > 0} ⊂ H. Notice that H 0 + ⊂ H ϕ ++ .
Hence, if the semigroup e tL preserves H 0 + , i.e. it maps H 0 + into itself, then A ϕ ++ (x 0 ) is not empty when x 0 ∈ H 0 + , as the null control c ≡ 0 belongs to it. Moreover, if also N is positivity preserving, we have the following monotonicity property:

c 1 ≤ c 2 =⇒ x x 0 ,c 1 ≥ x x 0 ,c 2 .
In particular

(2.5) c 1 ≤ c 2 , c 2 ∈ A ϕ ++ (x 0 ) =⇒ c 1 ∈ A ϕ ++ (x 0 ).
So, given x 0 ∈ H ϕ ++ , we are interested in the following optimal control problem:

(P ϕ ) Maximize J over the set A ϕ ++ (x 0 ), whose value function is

V ϕ (x 0 ) := sup c∈A ϕ ++ (x 0 ) J (c).
Note that we cannot say ex ante that V ϕ is finite. Sufficient conditions for finiteness will be provided later.

Remark 2.4. Problem (P ϕ ) is the most natural one from the mathematical point of view when ϕ is chosen suitably, in the sense that it admits an explicit solution. However, we anticipate that the meaningful problem from the economic point of view would be the (more difficult) one:

(P + ) Maximize J over the set A + (x 0 )
whose value function is

V + (x 0 ) := sup c∈A + (x 0 ) J (c).
where

A + (x 0 ) := {c ∈ L 1 loc (R + ; H + ) : x x 0 ,c (t) ∈ H 0 + for a.e. t ≥ 0}. Note that H 0 + ⊂ H ϕ ++ : hence, for x 0 ∈ H 0 + , A + (x 0 ) ⊆ A ϕ ++ (x 0 ) and, con- sequently V ϕ (x 0 ) ≥ V + (x 0 ). Moreover, if ĉ ∈ A ϕ ++ (x 0
) is optimal for (P ϕ ) and belongs to A + (x 0 ), then it is also clearly optimal for (P + ). In the illustration of the results, we will just test numerically, ex post, that the optimal control ĉ ∈ A ϕ ++ (x 0 ) also belongs to A + (x 0 ).

The HJB equation and the verification theorem

In this section we provide a verification theorem for the problem and, accordingly, the solution in a special case. The results will be used in the next section to treat our motivating economic application.

The Hamilton-Jacobi-Bellman (HJB) equation associated to the optimal control problem (P ϕ ) is

(3.1) ρv(x) = Lx, ∇v(x) + H(∇v(x)), x ∈ H ϕ ++ , where H(q) := sup z∈H + H CV (q; z), q ∈ H,
and H CV (q; z) := U(z) -N z, q , z ∈ H + , q ∈ H. Notice that the above supremum may not be finite in general. A sufficient condition for the finiteness is: U(z) ≤ C|z| α for some C > 0 and α ∈ (0, 1), N * preserving H ++ , and q ∈ H ++ . These will be used later.

Definition 3.1. We call classical solution to (3.1) (on H ϕ ++ ) a function v ∈ C 1 (H ϕ ++ ; R) such that ∇v ∈ C(H ϕ ++ ; D(L * )) and such that 1 ρv(x) = x, L * ∇v(x) + H(∇v(x)), ∀x ∈ H ϕ ++ .
Now we turn to state our Verification Theorem. Typically, to prove such theorem for infinite horizon problems, a condition on the solution v computed on the admissible trajectories when t → +∞ is needed. This is exactly the analogous of the so-called transversality condition arising in the maximum principle approach. The condition that we shall use is

(3.2) lim t→+∞ e -ρt v(x x 0 ,c (t)) = 0, ∀c ∈ A ϕ ++ (x 0 ). Theorem 3.2 (Verification). Let v be a classical solution to (3.1) in H ϕ ++ , let x 0 ∈ H ϕ ++ and let (3.2) hold. Then: (i) v(x 0 ) ≥ V ϕ (x 0 ); (ii) if, moreover, there exists ĉ ∈ A ϕ ++ (x 0 ) such that (3.
2) holds and

(3.3) N * ∇v(x x 0 ,ĉ (s)) ∈ D + U(ĉ(s)) for a.e. s ≥ 0,
where

D + U denotes the superdifferential of U, then v(x 0 ) = V ϕ (x 0 )
and ĉ is optimal for (P ϕ ) starting at x 0 , i.e.

J (ĉ) = V ϕ (x 0 ). Proof. (i) Let c ∈ A ϕ ++ (x 0 )
. By chain's rule in infinite dimension, we have, for every t ≥ 0,

d dt [e -ρt v(x x 0 ,c (t)))]
= e -ρt -ρv(x x 0 ,c (t)) + x x 0 ,c (t), L * ∇v(x x 0 ,c (t)) -N c(t), ∇v(x x 0 ,c (t)) .

Now we add and subtract e -ρt U(c(t)) to the right hand side, use the fact that v solves HJB, and integrate over [0, t]. We get, for every t ≥ 0,

e -ρt v(x x 0 ,c (t)) + t 0 e -ρs U(c(s))ds = v(x 0 ) + t 0 e -ρs -H(∇v(x x 0 ,c (s)) + H CV (∇v(x x 0 ,c (s); c(s)) ds,
Observe that, since U is concave (and hence sublinear from above), and c(•) ∈ L 1 loc (R + ; H + ) then both sides of the above inequality are finite for every t ≥ 0. Now, rearranging the terms and taking into account the definition of H, we get, for every t ≥ 0, (3.4) v(x 0 ) ≥ e -ρt v(x x 0 ,c (t)) + t 0 e -ρs U(c(s))ds 1 This equality, in particular, implies that H(∇v(x)) is finite for every x ∈ H ϕ ++ .

-with equality if c = ĉ verifies (3. Hence, passing (3.4) to the lim t→+∞ and using (3.2), we conclude

v(x 0 ) ≥ J (c)
Then, by definition of V ϕ and since c ∈ A ϕ ++ (x 0 ) was arbitrary, we immediately get the claim.

(ii) Notice that, by concavity of U, (3.3) is equivalent to

(3.6) ĉ(s) ∈ argmax z∈H ϕ ++ {U(z) -N z, ∇v(x x 0 ,ĉ (s))
}, ∀s ≥ 0, the usual closed loop condition for optimality. Hence, for c = ĉ we have equality in (3.4). Hence, passing to the lim t→+∞ and using (3.2) and (3.5), we get the equality v(x 0 ) = J (ĉ)

Since J (ĉ) ≤ V ϕ (x 0 ), combining with part (i), the claim follows.

A solution of the HJB equation

In this section we further specify the model to deal with an explicit solution. We consider the following assumption. (i) There exists an eigenvector b 0 ∈ H ++ for L * : D(L * ) ⊆ H → H with eigenvalue λ 0 ∈ R. Without loss of generality we assume that

|b 0 | H = 1. (ii) ρ > λ 0 (1 -γ). (iii) [N z](θ) = n(θ)z(θ), where n ∈ L ∞ (D, D, µ; (0, +∞)). (iv) u(θ, ξ) = ξ 1-γ 1 -γ f (θ), where γ ∈ (0, 1)∪(1, ∞) and f ∈ L ∞ (D, D, µ; R + ). (v) D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) < ∞, D f (θ) n(θ)b 0 (θ) p/γ µ(dθ) < ∞.
Notice that Assumptions 4.1(iv) implies Assumption 2.2. Also notice that, in general, b 0 may be not unique. A sufficient condition for the uniqueness of b 0 is that L * is a diagonal operator with respect to a given orthonormal basis in H. This will be the case in Section 6.

4.1) v(x) = α x, b 0 1-γ 1 -γ , x ∈ H b 0 ++ ,
where

(4.2) α = γ γ   D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) ρ -λ 0 (1 -γ)   γ is a classical solution to (3.1) on H b 0 ++ . Proof. Notice first that U is Fréchet differentiable in H b 0 ++ and [∇U(z)](θ) = f (θ)z(θ) -γ , z ∈ H b 0 ++ .

Hence, by straightforward computations

H(q) = γ 1 -γ D f (θ) 1 γ (n(θ)q(θ)) γ-1 γ µ(dθ), q ∈ H ++ ,
with optimizer

(4.3) ẑ(q)(θ) = argmax z∈H + U(z) -N z, q = f (θ) n(θ)q(θ) 1 γ . Moreover ∇v(x) = α x, b 0 -γ b 0 .
Plugging these expression into (3.1) and dividing all terms by x,b 0 1-γ 1-γ , we get the following algebraic equation in

α ∈ R ρα = λ 0 (1 -γ)α + γα γ-1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ),
which has a unique positive solution provided by (4.2).

The solution of the control problem

In order to produce an optimal control starting at x 0 ∈ H b 0 ++ , we study the closed loop equation associated to the solution v and to the candidate optimal feedback map (4.3) under Assumption 4.1:

(5.1)

x (t) = Lx(t) -N Φx(t), x(0) = x 0 ,
where Φ : H → H is the bounded linear positive operator

(5.2) [Φz](θ) = f (θ) αn(θ)b 0 (θ) 1 γ z, b 0 .
This linear equation admits a unique mild solution x, which is also a weak solution. In particular, testing against b 0 ∈ D(L * ) and taking into account the definition of α, it holds

d dt x(t), b 0 = x(t), L * b 0 -N Φx(t), b 0 = λ 0 -α -1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) x(t), b 0 .
Hence, taking int account the definition of α, providing

(5.3) α -1 γ D f (θ) 1 γ (n(θ)b 0 (θ)) γ-1 γ µ(dθ) = ρ -λ 0 (1 -γ) γ , we get d dt x(t), b 0 = g x(t), b 0 , i.e. x(t), b 0 = x 0 , b 0 e gt , t ≥ 0, where (5.4
ĉ(t, θ) := [Φx(t)](θ) = f (θ) αn(θ)b 0 (θ) 1 γ
x 0 , b 0 e gt , belongs to A b 0 ++ (x 0 ) and is optimal for (P b 0 ) starting at x 0 . Moreover, v(x 0 ) = V (x 0 ).

Proof. We want to apply Theorem 3.2. First, notice that, using the concept of weak solution (2.3) with ψ = b 0 , we have

d dt x x 0 ,c (t), b 0 = λ 0 x x 0 ,c (t), b 0 -N c(s), b 0 ∀c ∈ A b 0 ++ (x 0 ), i.e. x x 0 ,c (t), b 0 = x 0 , b 0 e λ 0 t - t 0 e λ 0 (t-s) N c(s), b 0 ∀c ∈ A b 0 ++ (x 0 ).
Since both n and b 0 are nonnegative (Assumption 4.1(i) and (iii)), we have

0 ≤ x x 0 ,c (t), b 0 ≤ x 0 , b 0 e λ 0 t ∀c ∈ A b 0 ++ (x 0 ). Hence, for every c ∈ A b 0 ++ (x 0 ), 0 ≤ e -ρt v(x x 0 ,c (t)) = αe -ρt x x 0 ,c (t), b 0 1-γ 1 -γ ≤ α 1 -γ x 0 , b 0 e -(ρ-λ 0 (1-γ))t .
Therefore, using Assumption 4.1(ii), we see that (3.2) hold. This allows to apply part (i) of Theorem 3.2.

Let us now turn to part (ii) of Theorem 3.2, i.e. to prove the optimality of ĉ. By construction x x 0 ,ĉ = x. Hence, by (5.5) and Assumption 4.1(v), we obtain ĉ ∈ A b 0 ++ (x 0 ). Moreover, again by construction, ĉ verifies the optimality condition (3.6). So, Theorem 3.2(ii) applies and the proof is complete.

Assumption 5.2. L * : H → H admits an orthonormal basis of eigenvectors, i.e. there exists an orthomormal basis {b n } n∈N of H such that

L * b n = λ n b n ∀n ∈ N. Moreover, (5.6) λ k < g, ∀k ∈ N \ {0}.
Theorem 5.3. Let Assumption 4.1 and 5.2 hold, and let x 0 ∈ H b 0 ++ . Then, setting xg := e -gt x(t), we have the convergence in H

xg (t) → x 0 , b 0 ∞ k=0 q k b k , as t → ∞,
where

q 0 := 1, q k := ζ k λ k -g ,
where

ζ k := f (•) αn(•)b 0 (•) 1 γ , b k
Moreover, the speed of convergence is exponential of order

s := sup k∈N\{0} {λ k -g}.
Proof. The equation for xg is (5.7)

x g (t) = (L -g)x g (t) -N Φx g (t).

Let

x(k) g (t) = xg (t), b k , so that, by Fourier series expansion

x(k) g (t) = ∞ k=0 x(k) g (t)b k .
From the fact that xg is also a weak solution to (5.7), we get the equations

d dt x(k) g (t) = (λ k -g)x (k) g (t) -x 0 , b 0 ζ k , k ∈ N.
We already know that, by definition of g,

x(0) g (t) ≡ x 0 , b 0 . For the others term, when k ∈ N \ {0}, we have

x(k) g (t) = x 0 , b k e (λ k -g)t + x 0 , b 0 ζ k 1 -e (λ k -g)t λ k -g .
The claim follows by (5.6).

The economic problem

We apply the results of the previous section to an economic problem by taking the following specifications of the general framework above:

(a) D = S 1 := {ξ ∈ R 2 : |ξ| = 1} ∼ = R/Z; S 1 is topologically identified
with [0, 2π] ⊂ R when the extreme of the latter interval are identified; similarly, functions on S 1 are identified with 2π-perodic functions on R; (b) µ = Hausdorff measure on the Borel σ-algebra of S 1 ; i.e., through the identification S 1 ∼ = R/Z, µ =Lebesgue measure on (0, 2π); (c) L = σ d 2 dθ 2 + A, where σ, A > 0 are constant; accordingly, the integration with respect to this measure will be simply denoted by dθ; (d) f (θ) = n(θ) β , with β ≥ 0.

In this case Assumption 2.1 is verified L = L * and this operator admits a spectral decomposition on the separable Hilbert space L 2 (S 1 , dθ; R). Moreover, there is a doubly indexed sequence of eigenvectors and eigenvalues of L = L * which is explicit. Indeed,

b 0 = (2π) -1/2 1 S 1 , λ 0 = A, and      b (1) k (θ) = π -1/2 sin(kθ), b (2)
k (θ) = π -1/2 cos(kθ), λ k = A -σk 2 , ∀θ ∈ S 1 , ∀k ∈ N \ {0}.
This formal frame fits quite well optimal growth models with AK production function, one-dimensional geography and capital diffusion as in Boucekkine et al. [START_REF] Boucekkine | Spatial dynamics and convergence: The spatial AK model[END_REF][START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]. Indeed, calling K(t, •) = x(t) according to the usual notation for capital in AK models, formally the problem becomes:

V (K 0 ) := sup A(K 0 ) ∞ 0 e -ρt S 1 c(t, θ) 1-γ 1 -γ n(θ) β dθ dt,
where ρ > 0, γ ∈ (0, 1), β ≥ 0,

A(K 0 ) := c ∈ L 1 loc ([0, +∞); L 2 (S 1 , dθ; R + )) : K(t, θ) ≥ 0 and K(t, θ) ≡ 0 for a.e. (t, θ) ∈ R + × S 1 , under the state PDE-constraint (6.1)    ∂K ∂t (t, θ) = σ ∂ 2 K ∂θ 2 (t, θ) + AK(t, θ) -n(θ)c(t, θ), (t, θ) ∈ R + × S 1 , K(0, θ) = K 0 (θ), θ ∈ S 1 ,
A few comments on the economic problem are needed at this stage. First of all, let us comment on the objective function,

∞ 0 e -ρt S 1 c(t, θ) 1-γ 1 -γ n(θ) β dθ dt,
which is indeed the social welfare function to be maximized by the central planner. This generalizes the social welfare function adopted by Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], which is itself an extension of Boucekkine et al. [START_REF] Boucekkine | Spatial dynamics and convergence: The spatial AK model[END_REF]. In the latter, n(θ) ≡ 1 and in the former β = 1. In this case, the planner sums all the utilities of all the individuals in location θ at time t. This corresponds to total utilitarianism or equivalently to the Benthamite case. When β = 0, one gets average utilitarianism or the Millian social welfare function: the planner do not consider all the individuals but only the average. In between, β ∈ (0 , 1), we get a continuum of intermediate configurations, we shall refer to this case as imperfect altruism: roughly speaking, as β increases, the planner takes more closely into account the welfare of his population. 2 Two other parameters of the social welfare functions are interesting to comment briefly on. Parameter ρ may be interpreted as the time discounting factor of the planner: the bigger ρ, the more weight will be given to the present (or to the current generations) by the planner. More interestingly, parameter γ has a double role in our spatiotemporal frame: it captures on one hand the cost of intertemporal consumption substitution faced by individuals, as in the standard non-spatial settings, but it also indicates the degree of aversion to inequality of the planner on the other. Thus one has therefore to expect that as γ increases, the less unequal will be the consumption distribution over space. We shall check this in the last section devoted to numerical illustration.

Finally the state equation (6.1) gives the spatiotemporal dynamics of capital. σ ∂ 2 K ∂θ 2 (t, θ) is the diffusion term of the equation, it depicts the dynamics of capital through space with σ > 0 the diffusion speed. The rest of terms are obvious. In particular n(θ)c(t, θ) is the so-called dilution effect of demographic size on capital accumulation. Notice that in this economy goods are produced and consumed locally, only capital moves cross locations (or more precisely, capital is moved by the planner according to the PDE above in line with Isard and Liossatos [START_REF] Isard | Liossatos Spatial Dynamics and Optimal Space-time Development[END_REF]).

Next, assuming that (A1)

S 1 n(θ) β+γ-1 γ dθ < ∞, S 1 n(θ) 2(β-1) γ dθ < ∞, (A2) ρ > A(1 -γ)
, all the assumptions of the previous sections are verified. By expliciting the result of Theorem 5.1 with these specifications, we get the following. Theorem 6.1. Let (A1)-(A2) hold and assume that the solution K to the linear integro-PDE

(6.2)    ∂K ∂t (t, θ) = σ ∂ 2 K ∂θ 2 (t, θ) + AK(t, θ) -α -1/γ (2π) -1/2 S 1 K(t, θ)dθ n(θ) β+γ-1 γ , K(0, θ) = K 0 (θ)
where t ∈ R + , θ ∈ S 1 , is such that K ≥ 0 and K = 0. Then, the following claims hold true.

(i) The value function is explicitly given by

V (K 0 ) = γ ρ -λ 0 (1 -σ) γ S 1 n(θ) β+γ-1 γ dθ γ (2π) -1/2 1 -γ S 1 K 0 (θ)dθ) 1-γ .
(ii) The control

ĉ(t, θ) = α -1/γ (2π) -1/2 S 1 K 0 (θ)dθ n(θ) β-1 γ e gt ,
where g := A-ρ γ is optimal and the corresponding optimal capital is K; g is therefore the optimal growth rate of the economy. (iii) If, moreover, A -σ < g and

∞ k=1 (ζ (1) k ) 2 + (ζ (2) k ) 2 < ∞, where ζ (2) k := ρ -A(1 -γ) γ S 1 n(θ) β+γ-1 γ dθ -1 S 1 n(θ) β-1 γ cos(kθ)dθ ζ (1)
k := ρ -A(1 -γ) γ S 1 n(θ) β+γ-1 γ dθ -1 S 1 n(θ) β-1 γ sin(kθ)dθ,
then the detrended optimal capital Kg (t, •) := e -gt K(t, θ)

converges in L 2 (S 1 , dθ; R + ) to the function K∞ g (θ) := (2π) -1/2 S 1 K 0 (θ)dθ 1 + ∞ k=1 ζ (1)
k sin(kθ) + ζ (2)
k cos(kθ) as e -(g-A+σ)t .

A few comments are in order here. Of course, since we are dealing with an AK model, growth is endogenous and all variables grow at exponential rates. The growth rates are not location-dependent. This is not surprising since technology (though parameter A) is the same everywhere, and so are individual preferences. Second, as in Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], we are able to identify the optimal stationary spatial distributions in closed-form and the corresponding convergence speeds, which certainly speaks about the flexibility of the analytical method developed. Third, and more importantly, β plays a central role in the shape of these distributions, in particular for consumption per capita. Indeed, this distribution is uniform if and only if the social welfare function is Benthamite (β = 1). This is an important property which was not of course reachable in the previous contributions to this area as they all rely on total utilitarianism. The next section goes deeper into this question via numerical exploration.

Some numerical illustrations

We shall now explore numerically some of the salient properties of the optimal long-term distributions theoretically uncovered above. We shall in particular, highlight the implications of two key parameters: the aversion to inequality parameter γ, and the altruism parameter β. To make the exposition more focused on the novelties brought by our generalization of social welfare, we shall concentrate on consumption, which is the control variable and the sole measure of welfare in our model. It can be shown that just as in Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], long-term total and per-capita capital, production and investments are lower in the more populated areas. This has a simple explanation: to guarantee a reasonable level of consumption to everybody, the planner needs to maintain an higher level of aggregate consumption in more densely populated areas leading to lower investment at the same locations. It follows that in this model where the optimal spatiotemporal decisions are only driven by demographic heterogeneity across space, and not by technology discrepancy, the key variable is consumption and the determinants behind (notably, aversion to inequality and altruism). 3 To start our numerical exercises, we need a benchmark (reasonable) calibration. Concretely, we choose: ρ = 0.04, γ between 0.6 and 0.9, β between 0 and 1 and A = 0.08. The value of ρ (the discount rate parameter) is chosen consistently with the data from Lopez [START_REF] Lopez | The social discount rate : estimates for nine Latin American countries[END_REF]. Parameter γ, also measuring the inverse of the intertemporal substitution with the CRRA specification of preferences, can be calibrated using relative risk aversion consistently with 3 We could have followed Allen and Arkolakis [START_REF] Allen | Trade and the Topography of the Spatial Economy[END_REF] and assume that the size of population drives productivity at any location. In other words, we could have assumed populationbased productivity heterogeneity. We don't to that to single out the pure population effect in our generalized context. the values found in individual choice experiments (see for example Cubitt et al., 2001 and Tversky and Kahneman [START_REF] Tversky | Advances in prospect theory: cumulative representation of uncertainty[END_REF]). The value of A is set to generate long-term growth rates of 4% to 6%, which are consistent with those observed in developing countries (see e.g. World Bank Group [23]). As to the exogenous density distribution, we assume that we have a demographic center as depicted in Figure 1: Because of the neoclassical dilution effect, the computed stationary distributions will always deliver that optimal consumption per capita goes down with population size over space except in the Benthamite case, which therefore features a case of egalitarianism. More interesting insights can be gained from the numerical exercises.

The implications of the aversion to inequality parameter, γ. Here we fix the altruism parameter β to 0, that is we stick to the Millian social welfare function. Needless to say, since we are concerned with consumption per capital, the Benthamite case, β = 1, as studied in Boucekkine et al. [START_REF] Boucekkine | Spatial dynamics and convergence: The spatial AK model[END_REF][START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF] is irrelevant as by the theorem just above, this is a case where consumption is uniform across space, independently of γ. Figure 2 delivers the stationary distributions of consumption per capita detrended for γ = 0.6, 0.75, 0.9. First, as γ increases, consumption is in average bigger. This is due to the fact that, as outlined in the previous section, γ also measures the cost of intertemporal substitution in consumption: when it's high, savings and investment are lower and consumption is higher. But in our spatiotemporal frame, γ also measures the planner's aversion to inequality. One can check that as γ rises, consumption distributions becomes less unequal. A rough and quick verification is to compute the ratio between the periphery plateau consumption value and its value at the trough of the distribution: it decreases steadily. 4 Figure 2 The implications of the altruism parameter, β. Figure 3 reports the stationary spatial distribution of detrended consumption capita. For this exercise, we fix γ = 0.8, and we play on the the altruism parameter. Precisely, we compute the distributions for the Benthamite (β = 1), Millian (β = 0) and an imperfect altruism (β = 0.5) cases. Not surprisingly, the optimal stationary distribution of consumption per capita is uniform. More interestingly, the Millian case (β = 0) delivers by far the largest spatial inequality in consumption while imperfect altruism displays intermediate results from this point of view. Indeed, the ratio between the periphery plateau consumption value and its value at the trough of the distribution is larger than 2, which exceeds the ratio between the corresponding population sizes. That is to say at low enough values of the altruism parameter β, the optimal spatiotemporal dynamics amplify the neoclassical dilution effect, even in the long-run! Including this parameter into the analysis seems therefore highly interesting from the normative point of view, and probably also from a more positive perspective in contexts of regional particularism.

Conclusion

In this paper, we have generalized the social welfare function typically considered in the recent literature on spatiotemporal growth with capital diffusion. Adapting the dynamic programming method in infinite dimension 4 In our calibrated example, this ratio move from 2.75 at γ = 0.6, to 1.85 when γ = 0.75 and finally it is equal to 0.75 for γ = 0.9. Empirical standard deviations deliver the same ranking of course. used in Boucekkine et al. [START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF], we have been able to solve in closed-form for the optimal controls and for the corresponding spatial stationary distributions as well, either in the short or in the long-run. We have found several interesting results. In particular, we prove that the Benthamite case is the unique one for which the optimal stationary detrended consumption spatial distribution is uniform. Interestingly enough, we also find that as the social welfare function gets closer to the Millian case, the optimal spatiotemporal dynamics amplify the typical neoclassical dilution population size effect, even in the long-run.

Further extensions with space-dependent altruism featuring regional particularism are worth exploring. We could have also studied much more in detail the interplay between altruism and aversion to inequality (that is between β and γ). We leave it for future work.

e

  -ρs U(c(s))ds = ∞ 0 e -ρs U(c(s))ds =: J (c).

  Assumption 4.1.

Proposition 4 . 2 .

 42 Let Assumption 4.1 hold. Then

(

Theorem 5 . 1 .

 51 Let Assumption 4.1 hold, let x 0 ∈ H b 0 ++ , and let v be the solution to (3.1) given as in Proposition 4.2. The control

Figure 1

 1 Figure 1

Figure 3

 3 Figure 3

  3) (equivalently, (3.6)) s-a.e. on [0, t]. Since U is nonnegative (Assumption 2.2), by monotone convergence theorem w have then it must be

  (R. Boucekkine) Aix-Marseille University, CNRS, EHESS, Centrale Marseille, AMSE. IRES corresponding member Email address: raouf.boucekkine@univ-amu.fr (G. Fabbri) Univ. Grenoble Alpes, CNRS, INRA, Grenoble INP, GAEL -CS 40700 -38058 Grenoble CEDEH 9, France Email address: giorgio.fabbri@univ-grenoble-alpes.fr (S. Federico) Università degli Studi di Siena, Dipartimento di Economia Politica e Statistica. Piazza San Francesco 7-8, 53100 Siena, Italy Email address: salvatore.federico@unisi.it (F. Gozzi) Dipartimento di Economia e Finanza, Libera Università degli Studi Sociali Guido Carli, Roma Email address: fgozzi@unisi.it

A natural extension is to consider that β is a function of θ, which is a direct way to model regional or local particularism. We abstract away from this in this paper.

Acknowledgements

The work of Giorgio Fabbri is supported by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02) and in that of the center of excellence LABEX MME-DII (ANR-11-LABX-0023-01).