Rethinking optimal control theory in resource economics

By Serguei Kaniovski, Economist with the Austrian Institute of Economic Research (WIFO)

Serguei Kaniovski and colleagues from IIASA and the Steklov Mathematical Institute of the Russian Academy of Sciences revisited a classic growth model in resource economics using recent advances in optimal control theory.

The late 1960s and early 1970s gave rise to Doomsday Models that predicted a collapse of Western Civilization under the pressure of over-population and environmental pollution. The very influential 1972 Club of Rome’s report on the “Limits to Growth” painted a gloomy picture, sparking an ongoing debate. One question was whether the scarcity of natural resources like fossil fuels would limit growth and cause a substantial decline in people’s standard of living.

The Doomsday reasoning was met with doubt by the economists of that time, leading the future Nobel Prize laureate and growth theorist, Robert Solow, to state that “the various Doomsday Models are worthless as science and as guides to public policy“. In a combined effort, economists developed a class of growth models with resource constraints. The conclusions they reached using the Dasgupta-Heal-Solow-Stiglitz (DHSS) modeling framework offered a more optimistic outlook.

© Kantver | Dreamstime.com

Economic applications have been well ahead of the mathematical theory used for identifying optimal economic policies, leaving some model solutions unexposed and some technical issues unsettled. The theory that allows us to identify optimal policies and describe the model dynamics was originally developed in the 1950s for engineering applications but has since become the main tool for analyzing economic growth models. These models however contain many features that are not standard to optimal control theory – a subfield of mathematics that deals with the control of continuously operating dynamic systems – which makes a fully rigorous analysis difficult. The key theoretical challenges are infinite planning horizons and nonstandard control constraints.

In our latest paper we offer a complete and rigorous analysis of the welfare-maximizing investment and depletion policies in the DHSS model with capital depreciation and arbitrary (decreasing, constant, and increasing) returns to scale. The investment policy specifies the portion of the final output to be invested in capital. A depletion policy says how fast a finite stock of exhaustible resources should be used. We prove the existence of a solution and characterize the behavior of solutions for all combinations of the model parameters using necessary rather than sufficient (Arrow’s theorem) optimality conditions.

In the main case of decreasing, constant, or weakly increasing returns to scale, the optimal investment and depletion policies converge to a constant share of output invested in capital and a constant rate of depletion of the natural resource. The optimal investment ratio decreases with the longevity of capital and impatience. The relationship between the optimal investment ratio and the output elasticity of produced capital is ambiguous. The performed analytical analysis identifies those relationships among model parameters that are critical to the optimal dynamics. In this, it differs from more conventional scenario-based approaches. From a practical point of view, application of the model to real data could be helpful for evaluating actual depletion and investment policies.

Strongly increasing returns to scale make it optimal to deplete the resource without investing in produced capital. Whether a zero-investment strategy is followed from the outset, from an instant of time, or asymptotically will depend on the sizes of the capital and resource stocks. In some special cases of increasing returns, welfare-maximizing investment and extraction policies may not exist under strong scale effects in resource use. This occurs when an initial stock of capital is small relative to the initial resource stock. It implies that it would have been impossible to formulate a welfare-maximizing policy in the early history of humanity, when produced capital was scarce and resources were abundant.

Reference

Aseev S, Besov K, & Kaniovski S (2019). Optimal Policies in the Dasgupta—Heal—Solow—Stiglitz Model under Nonconstant Returns to Scale. Proceedings of the Steklov Institute of Mathematics 304 (1): 74-109. [pure.iiasa.ac.at/15946]

Note: This article gives the views of the author, and not the position of the Nexus blog, nor of the International Institute for Applied Systems Analysis.

Running global models in a castle in Europe

By Matt Cooper, PhD student at the Department of Geographical Sciences, University of Maryland, and 2018 winner of the IIASA Peccei Award

I never pictured myself working in Europe.  I have always been an eager traveler, and I spent many years living, working and doing fieldwork in Africa and Asia before starting my PhD.  I was interested in topics like international development, environmental conservation, public health, and smallholder agriculture. These interests led me to my MA research in Mali, working for an NGO in Nairobi, and to helping found a National Park in the Philippines.  But Europe seemed like a remote possibility.  That was at least until fall 2017, when I was looking for opportunities to get abroad and gain some research experience for the following summer.  I was worried that I wouldn’t find many opportunities, because my PhD research was different from what I had previously done.  Rather than interviewing farmers or measuring trees in the field myself, I was running global models using data from satellites and other projects.  Since most funding for PhD students is for fieldwork, I wasn’t sure what kind of opportunities I would find.  However, luckily, I heard about an interesting opportunity called the Young Scientists Summer Program (YSSP) at IIASA, and I decided to apply.

Participating in the YSSP turned out to be a great experience, both personally and professionally.  Vienna is a wonderful city to live in, and I quickly made friends with my fellow YSSPers.  Every weekend was filled with trips to the Alps or to nearby countries, and IIASA offers all sorts of activities during the week, from cultural festivals to triathlons.  I also received very helpful advice and research instruction from my supervisors at IIASA, who brought a wealth of experience to my research topic.  It felt very much as if I had found my kind of people among the international PhD students and academics at IIASA.  Freed from the distractions of teaching, I was also able to focus 100% on my research and I conducted the largest-ever analysis of drought and child malnutrition.

© Matt Cooper

Now, I am very grateful to have another summer at IIASA coming up, thanks to the Peccei Award. I will again focus on the impact climate shocks like drought have on child health.  however, I will build on last year’s research by looking at future scenarios of climate change and economic development.  Will greater prosperity offset the impacts of severe droughts and flooding on children in developing countries?  Or does climate change pose a hazard that will offset the global health gains of the past few decades?  These are the questions that I hope to answer during the coming summer, where my research will benefit from many of the future scenarios already developed at IIASA.

I can’t think of a better research institute to conduct this kind of systemic, global research than IIASA, and I can’t picture a more enjoyable place to live for a summer than Vienna.

Note: This article gives the views of the author, and not the position of the Nexus blog, nor of the International Institute for Applied Systems Analysis.

Modeling forest fires: a burning issue

By Andrey Krasovskii, IIASA Ecosystems Services and Management Program

By 2090, the area burned by forest fires in the European Union could increase by 200% because of climate change.  However, preventive fires could keep that increase to below 50%. Improved firefighting response could provide additional protection against forest fires. These findings were the result of modeling work we did for the EU Mediation project  on projecting future burned areas and adaptation options in Europe.  When we talk about these results, people often want to know more about how our model works, what assumptions it makes, and how reliable it is.

 

Figure 1. The WildFire cLimate impacts and Adaptation Model (FLAM) schematic – estimation of expected burned area.

Figure 1. The WildFire cLimate impacts and Adaptation Model (FLAM) schematic – estimation of expected burned area.

The model is complex: every link in the schematic shown above represents a specific mathematical formula. These formulas have been developed by many researchers who studied how wildfire occurrence is related to climate, population, and biomass available for burning. Their results have been aggregated into mathematical relations and functions attempting to replicate real processes. The model code runs through the scheme with daily weather inputs in order to calculate the potential for fire ignition, spread, and burned areas. The model transforms spatial and intertemporal inputs into expected burned areas for 25km squares across the entirety of Europe. These squares can be summed up into geographic regions, e.g. countries, as well as burned areas can be aggregated over a given time period, e.g. 10 years.

It took days for our colleague Mirco Migliavacca to run the model during his work at the Joint Research Center of the European Commission. In fact, the scheme depicted in Figure 1 shows only a small piece of a larger picture reflecting the Community Land Model with the integrated fire module (CLM-AB), which he used. CLM-AB calculates all inputs in the indicated fire module, based on modeling processes in the global vegetation system. To speed up the running times for the case study focused on the wildfires in Europe, my colleague Nikolay Khabarov developed a standalone version of the fire model by decoupling the fire module from CLM-AB. When I joined the study, we had also found alternatives for input data, e.g. IIASA’s Global Forest Database, and implemented additional procedures in order to create our wildfire climate impacts and adaptation model (FLAM).

We used the historical data from satellite observations in order to validate modeling results. At the beginning many numerical experiments in CLM and FLAM did not give satisfactory results – there was either overestimation or underestimation of modeled burned areas compared to those reported in available datasets. One day a purely mathematical insight happened. We realized that in the fire algorithm implemented in FLAM, there is a parameter that can be factorized, mathematically speaking. This parameter, a probability of extinguishing a fire in a pixel in one day, was constant for Europe and set to 0.5. It became obvious that this parameter should vary with respect to a region. Factorization of this variable gave a possibility to avoid routine calculations, and use it for calibrating the model over a historical period. This can be done analytically by solving a corresponding polynomial equation. Analytical findings allowed us to introduce an effective calibration procedure and at the same time to estimate a firefighting efficiency on a country level. Further, using the advice of our colleagues Anatoly Shvidenko and Dmitry Schepaschenko, we have introduced adaptation options in the model, for example prescribed burnings, which firefighters use to reduce the fuel availability and, consequently, potential of a major fire.

(cc) US Bureau of Land Management

Prescribed burnings are one tool that can help prevent major wildfires. (cc) US Bureau of Land Management via Flickr

Once we had calibrated the model so that it adequately performed on the historical period (using historical climate data), we used climate scenarios to produce future projections. Currently, we are working on further improvements in modeling accuracy in annual burned areas by introducing additional regionally specific factors in the model. In the recent study published in the International Journal of Wildland Fire, we suggested improving the original model by modifying the fire probability function reflecting fuel moisture. This modification allows for a dramatic improvement of accuracy in modelled burned areas for a range of European countries.

Despite some success in modeling annual burned areas in Europe, we still have difficulties in predicting the extreme fires, in particular in some more arid and hence vulnerable regions such as Spain. However, we accept the challenge, because credible modeling results in terms of burned areas provide important information for assessing economic damages and CO2 emissions, due to climate and human activities. Our research has the potential to help society to realize these risks and undertake preventive measures. It also delivers an additional scientific value due to the fact, that fire risks must be included in forest management models.

I would like to thank all the study co-authors for their valuable contributions and efficient collaboration.

Reference
Krasovskii, A., Khabarov, N., Migliavacca, M., Kraxner, F. and Obersteiner, M. (2016) Regional aspects of modelling burned areas in Europe. International Journal of Wildland Fire. http://dx.doi.org/10.1071/WF15012 

Note: This article gives the views of the interviewee, and not the position of the Nexus blog, nor of the International Institute for Applied Systems Analysis.

The mathematics of love

By Sergio Rinaldi, IIASA Evolution and Ecology Program and Politecnico di Milano, Italy

Is it possible to predict how love stories develop, progress, and end using mathematical models? I have studied this question over the past 20 years with a group of researchers at IIASA and at the Politecnico di Milano, and as we show in our new book Modeling Love Dynamics (World Scientific, 2016), the answer is yes. The emerging message is that prediction is possible, if we can describe in formulas the way each individual reacts to the love and to the appeal of the partner.

Consider a standard love story, which develops like those described in a classical Hollywood movie such as Titanic. This story can be easily modeled, if one considers reasonably appealing individuals who increase their reaction with the partner’s love – so called secure individuals. Starting from the state of indifference, where the individuals are at their first encounter, their feelings continuously grow and tend toward a positive plateau.

Mala Powers & José Ferrer in Cyrano de Bergerac, 1950. Credit: Public Domain

Mala Powers and  José Ferrer in Cyrano de Bergerac, 1950. – Public Domain

Love stories become more intriguing when one individual is not particularly appealing, if not repelling, as in the fairy tale “Beauty and The Beast.” Indeed, in these cases, there exists also a second romantic regime, which is negative and can therefore entrain, in the long run, marital dissolution. In order to avoid that trap, people who are not very charming, or believe to be so, do all they can to look more attractive to the partner. At the first date, she wears her nicest dress and he shows up with his best fitting T-shirt. However, after a while, the bluffing can be interrupted, because the couple has entered the safe basin of attraction of the positive regime. Needless to say, the model also supports much more sophisticated behavioral strategies, like that described by Edmond Rostand in his “Cyrano de Bergerac,” the masterpiece of the French love literature.

Not all individuals are secure. Indeed, some people react less and less strongly when the love of the partner overcomes a certain threshold. These individuals, often very keen to flirtation, are incapable of becoming one with their partner. The model shows that couples composed of insecure individuals tend, with almost no exception, toward an unbalanced romantic regime in which the most insecure is only marginally involved and is therefore prone to break up the relationship at the first opportunity. This is why after just 20 minutes of the very long “Gone with the Wind,” when one realizes that Scarlett and Rhett are both insecure, the model can already predict the end of the film, where he quits her with the lapidary “Frankly, my dear, I don’t give a damn.” The same conclusion is expected if only one of the two individuals is insecure. This explains the numerous failures in the romantic life of some individuals, like the beautiful star Liz Taylor, who is described as very insecure in all her biographies, and went, indeed, through eight marriages.

Clark Gable and Vivien Leigh in

Clark Gable and Vivien Leigh in Gone with the Wind, 1939 – MGM Pictures | Public Domain

Mathematical models can also be used to interpret more complex romantic behaviors. Particularly important is the case of individuals who overestimate the appeal of the partners when they are more in love with them (like parents who have a biased view of the beauty of their own kids). Interestingly, if insecurity is also present, biased couples can have romantic regimes characterized by recurrent ups and downs. In other words, the theory says that bias and insecurity is an explosive mix that triggers turbulence in the life of a couple.

In the second part of the book we focus on  the effects of the social environment and to the consequences of extra-emotional compartments. In this context, our analysis of the 20-years long relationship between Laura and the famous Italian poet Francis Petrarch shows that poetic inspiration is an important destabilizing factor, responsible for transforming a quiet relationship into a turbulent one.

Finally, we studied triangular relationships, with emphasis on the effects of conflict and jealousy. In all these cases the dynamics of the feelings can be very wild, up to the point of being chaotic and, hence, unpredictable. When this occurs, the life of the couple becomes unsustainable, because painful periods of crisis can virtually start at any moment: a heavy permanent stress. The model can thus explain why the relationship is often interrupted, sometimes even tragically, as in the famous film by François Truffaut “Jules et Jim”, where Kathe’s suicide is perceived as a real relief.

CaptureMore information: Watch a video of Sergio Rinaldi’s talk at the 2015 Systems Analysis Conference.

Reference
Rinaldi S, Della Rossa F, Dercole F, Gragnani A, Landi P, (2015). Modeling Love Dynamics. World Scientific, Singapore [January 2016]  http://www.worldscientific.com/worldscibooks/10.1142/9656

 

Note: This article gives the views of the author, and not the position of the Nexus blog, nor of the International Institute for Applied Systems Analysis.

This summer in Moscow: Impressions from Moscow Summer Academy 2015

By M. Nazli Koseoglu, MSA 2015 Participant, School of Geosciences, Environmental Economics, Edinburgh University, Scotland

M. Nazli Koseoglu

M. Nazli Koseoglu

The Summer Academy on Economic Growth and Governance of Natural Resources took place at Lomonosov Moscow State University from 20th July to 1st August 2015.

As an environmental economist working on economic valuation and optimisation of water use, the academy was very interesting for me. Water management is a dynamic process and requires bringing perspectives and expertise from different disciplines together. Application of systems analysis enables us to combine aspects from various domains, come up with models that identify nonlinearities, project regime shifts, and tipping points in the management of water as well as other natural resources. Such projects require interdisciplinary collaboration and communicable results to inform policy. Scientists need to translate their results to a language accessible to the policymakers, in order for society to pick up on and capitalize on the research efforts. The MSA 2015 provided me with necessary training to go deeper into different modelling methodologies, and learn the concepts and principles of science for policy first-hand from IIASA scientists.

The reading list sent before the course gave me the impression that I would probably be the only environmental economist amongst a crowd of mathematical modellers. However, arriving in Moscow, I found that the MSA 2015 participants came from a broad range of backgrounds and countries at different stages of their careers in academia or policy. We all came  to learn and discuss the natural resource constraints to infinite economic growth on finite planet.

During lectures, the theoretical foundations of different mathematical approaches such as dynamical systems theory, optimal control theory and game theory were presented by leading scientists, such as Michael Ghil. Fundamentals of addressing challenges of natural resource management and comparing contemporary models of economic growth were also covered as central themes.

The course acknowledged the issues related with ecosystems services, public goods, inter-generational and international fairness, and public and common pool resource dynamics in the face of economic growth and resource constraints. The training underlined feedbacks between institutional dynamics and resource dynamics in complex social-ecological system and need for interdisciplinary and policy-relevant research, an important take-home message for next generation scientists.

Photos by M. Nazli Koseoglu

Photos by M. Nazli Koseoglu

What makes the MSA so special?
Apart from lectures, we had tutorials, a group project, poster and project presentation sessions, as well as interesting talks on IIASA activities by Margaret Goud-Collins and Elena Rovenskaya, and an inspiring session on the importance of finding the right mentor for a successful career by Prof Nøstbakken. The  MSA 2015 program had a good balance of theory and practice, which encouraged participants to be proactive and engaged.

I particularly liked the poster session. We presented our ongoing projects and received feedback from the lecturers and other participants. It was great to get comments and perspectives that I never thought of, and tips from senior researchers. In the late days of the academy we were assigned to prepare a group project on Artic systems which allowed us to put what we had learned at the lectures into practice and apply important topics outside our exact fields of study; in my case, these topics were petroleum economics and artic futures. I found the multi-disciplinary group work to be a great exercise for the development of my current study.

Attending the MSA 2015 provided useful training, both theoretical and practical, for understanding systems analysis approaches better. The host institution and organizing committee at Lomonosov Moscow State Univesrity provided impeccable hospitality, and the setting, in a landmark building in a landmark city, was a great perk. I received very constructive feedback, and made good connections around the world. I would recommend all early-career researchers in relevant fields to take this great opportunity next summer!

More information about MSA 2015

Participants in the MSA 2015. Photo Credit: M. Nazli Koseoglu

Participants in the MSA 2015. Photo Credit: M. Nazli Koseoglu

Note: This article gives the views of the author, and not the position of the Nexus blog, nor of the International Institute for Applied

Systems Analysis.