This paper is one of the outcomes of a joint research project by several experts from different working contexts. Can you summarize the main focus of this research project and of any related projects you are working on? Can you also explain why research collaboration on this area is important?

In recent years, I started to collaborate with a group of researchers that includes experts in applied mathematics, biology and economics. The overall focus of our research program is the interdisciplinary study of the impact of economic activities on the environment (and, consequently, on the wellbeing of populations) and of the strategies that regulators (e.g. governments) can adopt to face this issue, possibly through interacting with each other.

In particular, we look at the above-mentioned impact of economic activities from four different viewpoints:

I) the effect of “local” pollution (i.e. poisoning emissions directly affecting people and the environment);

II) the effect of “global” pollution (CO2 and other greenhouse gases which have indirect effects on populations through climate change);

III) the effect of biodiversity loss, given that our “homo sapiens” species has brought to extinction roughly 1/3 of all species in five centuries. This is what is now referred to as the sixth mass extinction.

IV) the effect of increased probability of pandemics such as COVID-19.

The key features of our approach are as follows:

– to formulate and study mathematical models for the above themes, using new advanced mathematical techniques that have recently been developed, in particular “Infinite Dimensional Optimal Control”, “Differential Games” and “Mean Field Games”;

– to simulate and test such models using real data such as those from the recent pandemics;

– to formulate suggestions for the policy makers, on the basis of the above study, e.g. on the effectiveness of climate agreements.

Such problems are currently studied by many researchers around the world. The main novelty of our approach is to put together some recent advances of applied mathematics with the latest knowledge in economics and biology.

Such a project can only be implemented with the contribution of scientists from different and complementary backgrounds (applied maths, biology, economics). Only in this way is it possible to be at the frontier of real knowledge and to exploit the connections from different perspectives.

Our paper is one of the first arising from this broad project and concerns point I) above, i.e. “local” pollution.

In your paper you study the joint determination of optimal investment and optimal depollution within a spatio-temporal framework in which pollution is transboundary and controlled at global level.

Starting from the specific regulator assumptions that production generates pollution – which is detrimental for the wellbeing of the population – and that pollution flows across space driven by a diffusion process, what are the main findings of your research? In particular, what are the key aspects to be considered by the regulator in order to adopt the optimal policy under the conditions you analyse?

Local pollution arising from production spreads throughout the territory over time. The regulator should find a production/depollution policy that harmonizes economic growth with the wellbeing of the population.

In our paper we study a mathematical model for such a situation. The main novelty, with respect to the previous literature, lies in the fact that we deal with a more realistic set-up

where the distribution of the pollution and of the population strongly vary across space: what we call the “geographical heterogeneity” of such variables.

What this means is that our model may permit a qualitative/quantitative prediction of the mid-term and long-term effects of the production/depollution strategies chosen by regulators, taking into account the key aspects of geographical and population heterogeneity. This may become a powerful policy-advising tool and could serve to avoid environmental/wellbeing problems such as the ones we observe in many places in Italy (e.g. Taranto).

More specifically, we model the time-space diffusion of the pollutants by using a classical second order Partial Differential Equation (PDE). The solution can be modified by the choices of the regulator: investments in production (e.g. new chemical plants, which increase the pollution) and investment in abatement (e.g. clean technologies) which can be carried out in different places, taking geographic heterogeneity into account.

The regulator chooses such investments with the aim of reaching the maximum possible welfare of the population. The model is quite flexible as it permits an implementation of a different spatial configuration of the region and different ways to measure the welfare. In the paper we provided some examples of simulations, but our framework could potentially be adapted to different regions and to various attitudes adopted by the regulators.

A companion paper examines what happens, in the same context, when a region of the planet (e.g. Europe) is divided into subregions (e.g. the actual nations) with different regulators who do not necessarily agree. We emphasize that in such a case the cooperative strategy (which would be the best one from a global viewpoint) may not be the one adopted by the regulators, leading to an increase in overall pollution, in particular at the borders of the nations (what we call the “border effect”).

From a qualitative perspective, I might just say two things that may seem obvious: firstly, that cooperative solutions in this context are always better than selfish ones; secondly, that it is crucial to have a real understanding of the consequences of the regulator’s decisions, to carefully take into account both the time and space dimension) when looking at this problem, hence moving, without hesitation, to exploit advanced applied math tools such as the ones recalled above.

 In your opinion, what are the basic skills that a researcher interested in studying and applying mathematical methods for Economics and Finance should have?

It depends to a great degree on the overall goal of the researcher. In any case, I would say that it is very important to have a strong basic knowledge in applied mathematics, in particular mathematical analysis, probability and stochastic processes, optimal control and game theory. Such knowledge can be attained, e.g. by taking a PhD in Applied Mathematics.

On the other hand, it is equally fundamental to have a strong desire to discuss and cooperate fruitfully with people from different backgrounds. This is not an easy task and takes time and energy to set up a common language; but it is certainly a powerful tool to make significant advancement in research on such complex problems.