<figure><img src="image_1.png" /></figure>

HAL
open science

# Macrodynamics and climate: reformulation

Gaël Giraud, Paul Valcke

► To cite this version:

Gaël Giraud, Paul Valcke. Macrodynamics and climate: reformulation. Oxford Open Economics, 2023, 2, 10.1093/ooec/odad007. hal-04872596

HAL Id: hal-04872596

https://hal.science/hal-04872596v1

Submitted on 8 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Macrodynamics and climate: reformulation

Gaël Giraud<fnref n="1" />,* and Paul Valcke<fnref n="2" />

## Abstract

Designing policy for global warming requires an integrated analysis of the interplay between the economy and the environment. The consensus is growing that, despite their dominance in the economics literature and their influence in public discussion and policymaking, the methodology employed so far by most Integrated Assessment Models (iams) 'rests on flawed foundations' (Stiglitz et al. 2016). This is particularly worrisome in the face of the immense risks and challenges of global warming and the radical changes in our economies that an effective response requires. This paper introduces an alternative paradigm, idee (Integrated Dynamic Environment-Economy), based on coupling a medium-size climate model with nonlinear, out-of-equilibrium, stock-flow-consistent macroeconomic dynamics in continuous time. Idee allows for multiple economic steady states, endogenous business cycles, endogenous growth, corporate default, and the short- and long-run assessment of various mitigation and adaptation policies. We argue that this approach is suitable for providing insights into managing the transition to net-zero emissions and coping with damages induced by the ecological crisis.

**Keywords:** climate change, extreme risk, market imperfections, climate policy, integrated assessment, stock-flow consistency.

## Introduction

When he said in September 2015: 'A wholesale re-assessment of prospects, as climate-related risks are reevaluated, could destabilize markets, spark a procyclical crystallization of losses and lead to a persistent tightening of financial conditions: a climate Minsky moment,' (A Minsky moment is a sudden collapse of financial market prices due to overinvestment in financial assets funded by debt, which leads to an unsustainable decoupling between the rapidly growing requirement of debt servicing and the flow of income stemming from the real economy, cf. (Minsky 1982).) Mark Carney, then governor of the Bank of England, reported a belief that is hardly disputed today, in financial circles.

The risks are manifold. On top of physical risks, there are transition risks. If we go too fast towards decarbonization, we may trouble several financial institutions that rely heavily on fossil fuel assets. (Giraud et al. 2021) underline a cornerstone of the climate risks, in that the first 11 banks in the eurozone hold €530 bn in fossil fuel assets, representing an average of 95% of each bank's equity. These assets' value would crumble to nearly nothing if oil, coal, and gas became stranded assets over the next few years. Fossil fuel assets are analogous to subprime credit derivatives during the Global Financial Crisis (gfc). Thirdly, there are also legal risks, with ngos filing suits against governments as the latter fail to take proper action to hold their climate commitments, such as the Paris Agreement. (At the time of writing, France's leading bank, BNP-Paribas, is the first bank in the world to be sued for its financing of fossil fuels. Https://www.reuters.com/article/bnp-litigation-climate-change-idAFL1N3530L4)

This paper focuses on the first type of climate risk: the physical impact of global warming on our economies and its potential

ability to destabilize global financial markets. We introduce a new family of models integrating climate and the economy. Section 2 first briefly overviews how conventional climate macroeconomics has approached climate risk in recent decades. High-lighting some critical shortages that emerged during this process, we conclude that the gross underestimation of physical risks by standard Integrated Assessment Models (iams) may lead the profession to the same kind of blindness that prevented many of us from anticipating the gfc. We introduce then in Section 3 complexity economics as an alternative path to studying out-of-equilibrium dynamics in systems with multiple equilibria, some of which are deeply inefficient. More precisely, we provide an account of the recent literature devoted to Lotka-Voltera dynamics and discuss their coupling with climate models. Such a combination is a promising alternative to conventional iams. Subsection 3.2 presents the backbone of any complex economic model: stock-flow consistency. Section 4 discusses the short-run Phillips curve in the context of such models, then introduces the seminal Goodwin (1967) model, which serves as a starting point for more advanced dynamical systems. We then discuss two extensions: the Van der Ploeg (1985) model and the Keen (1995) model. Section 5 introduces climate feedback on the previous model and its destabilization. Section 6 discusses climate models of higher complexity and recent work that aims to integrate macroeconomic models with models of intermediate complexity to answer the limitations of traditional iams. We introduce a new integrated model class, idee. In the concluding Section 7, we discuss the important change of paradigm provided by idee on climate policy and what new development it can bring.

<footnote n="1">¹McCourt School of Public Policy, Environmental Justice Program, Georgetown University, 37th and O Street NW, Washington, DC 20057, USA</footnote>

<footnote n="2">²Environmental Justice Program, Georgetown University, 37th and O Street NW, Washington, DC 20057, USA</footnote>

<footnote>*Corresponding address. Suite 391 Regents Hall, 37th and O Streets NW, Washington, DC 20057, USA. E-mail: gael.giraud1224@gmail.com</footnote>
# Shortcomings of the conventional macroeconomics of climate Integrated assessment models

Many macroeconomists' likely answer to Carney is that physical risks are not incredibly worrying in the first place. Despite the growing scientific understanding that nature and humans in the Anthropocene are intertwined, conventional macroeconomics falls short of correctly modeling human dependence and influence on our environment.

The shortcoming is manifest in the economics literature dedicated to climate change. The main stage stream of literature in environmental economics looks at the issue through the lens of Integrated Assessment Models (or AIMS).

AIMS are global models that combine simplified economic, energy/resources, and environmental models. On the environmental side, climate models vary in complexity, from highly stylized models focused on only a few variables without spatial heterogeneity (as in dice) to highly complex, fine-grained models of the physical climate systems based, among other things, on fluid dynamics and thermodynamics (Global Circulation Models, gcm). On the anthropic side, the economic model of most iams builds on some version of an optimal growth model with 'rational expectations' (Ramsey 1928), (Koopmans 1963), (Solow 1988). Strong assumptions are often made on exogenous growth and technical progress. On the energy and natural resource end, they typically do not account for the relative scarcity of several minerals (Vidal et al. 2017, Vidal et al. 2019), not mentioning the decline of eroi (Energy Return on Investment (Hall et al. 2001)), the accelerating erosion of biodiversity (Sage 2019), and the growing human appropriation of primary production of biomass (Kraussmann et al. 2013). Demography is often assumed to be exogenous despite the growing imbalance induced by aging. (Today, 10% of the world population is above 65 years; certain scenarios envision that this proportion may grow up to 25% before 2100.) The dependency of the climatic module on the economic sphere typically takes the form of a radiative forcing effect of greenhouse gases induced by economic activities, which affects the temperature of the layer of the atmosphere where such emissions stagnate. On the other hand, the dependence of the economic module on climate relies on a damage function, which translates a difference in average global temperature from the pre-industrial era to a percentage loss of global economic output.

IAMS were originally devised to estimate the optimal investment strategies in abatement costs. Some authors take a Pigouvian approach to translate these results into policy, viewing greenhouse gas as a negative externality responsible for markets' pareto-inefficiency. The main interest of an iam, then, is to enable the computation of the 'optimal pricing' of carbon that should internalize the negative externality into the market price system. The corresponding cost of carbon is referred to as the Social Cost of Carbon (scc), and can be implemented through taxation. To illustrate their methodology, consider the dice model, probably the most widely quoted iam (Nordhaus 1992), updated in (Barrage and Nordhaus 2023) with marginal changes of the normativity produced. The economic module in dice is based on the Ramsey-Cass-Koopmans optimization program (Koopmans 1963), (Cass 1965) and introduces natural capital along with industrial capital. The former can be 'saved' by investing in abatement at the latter's expense. On the consumption side, the model considers a representative household with a concave utility function and perfect foresight. On the production side, a representative firm produces a single good with a Cobb-Douglas technology. Crucially,

Total Factor Productivity increases exogenously and faster than the world population. This assumption naturally leads the social planner to favor postponing investment in abatements as the per-capita wealth of the economy increases, leading to lower marginal productivity of capital as time goes on (since capital is assumed to display decreasing returns to scale), hence, to a lower marginal disutility of investment in abatement technologies. Although central in multiple iams, such mathematical core has proven to be misleading in economic and climate projections (Ryzhenkov 2022), (Pottier 2014).

Moreover, as an intertemporal social planner's problem, dice relies on a social discount rate whose value significantly influences the 'optimal' production, consumption, and emissions trajectory. In the current context of global warming, in particular, the discount rate carries both philosophical and political weight and is, up to a point, a matter of ethical choice. According to Ramsey's formulation of the problem, the socially optimal discount rate, $\rho$, should be given by

$$ \rho = \delta_{\rho} + \sigma g, \quad (1) $$

where $\delta_{\rho} \ge 0$ is the pure rate of time preference, $\sigma$ the individual degree of relative risk aversion and $g$ the real growth rate of the economy. This choice has been the subject of much controversy beyond the initial quarrel between Nick Stern and William Nordhaus (Sterner and Persson 2008). Authors that favor very low discount rates—on the basis that future generations have the same rights in terms of environmental well-being or because ecosystems must be protected, either as a fundamental imperative or from a utilitarian perspective, so that $\delta_{\rho} = 0$ in either case—typically end up with trajectories with steeper investment in abatement and a more contained temperature rise. The Stern report (Stern 2008) is among these authors, choosing a discount rate of 0.01%.

dice chooses a much larger discount rate (With the notations of (1), $\sigma = 1$ and $\rho \sim 3 - 4.25\%$.) and concludes that temperature profiles reaching +3.5°C above pre-industrial levels would be 'socially optimal'—with an increase of +2°C in as early as 2050. These recommendations stand in sharp contrast with the climate science literature. To give but one example, a group of climate scientists recently broke a taboo by arguing that we should be able to speak openly about the possible extinction of humanity induced by global warming in the not-too-distant future (Kemp et al. 2022). There is a quasi-unanimity of the climate science community that a +3.5°C temperature anomaly would potentially lead to catastrophic environmental disruptions and large losses in welfare (Knutti et al. 2015).

Within the marginalist tradition, Mertens and Rubinchik (2017) show that the (relative) Utilitarian criterion (Relative utilitarianism is a Social Welfare Function (Dhillon and Mertens 1999) given by the uniform average of normalized utilities of citizens. The 'tour de force' lies in the normalization procedure, which allows axiomatically 'deduces' cardinal utilities from ordinal preferences.) unambiguously leads to rewriting the Ramsey equation (1) as follows:

$$ \rho = g. \quad (2) $$

In other words, risk aversion does not interfere with the economic problem of choosing the 'right' discount rate ($\sigma = 1$). As for the pure rate of time preference, $\delta_{\rho}$ should vanish. This implies that the controversy over the discount factor boils down to assessing the world economy's future real growth rate, $g$. However,
this is where we again come up against one of the severe limitations of the iam approach: in addition to the many discussions raging around the possibility of secular stagnation (Gordon 2011), it turns out that, at the long-run equilibrium of most conventional models, the growth rate of the economy is entirely exogenous—and usually given by the sum of population growth and labor productivity growth. In light of (2), this amounts to saying that the controversial choice of the discount rate is left to the arbitrariness of the calibration by us, economists, of the unobservable long-run exogenous growth rate of our models.(An objection to this criticism could go as follows: in (Nordhaus 2018), for instance, ρ is calibrated according to the marginal productivity of capital, as observed by current short run interest rates on financial markets. That the latter may equal capital marginal productivity, however, is true only under an important number of assumptions (decreasing returns of capital, equilibrium interest rate, the efficiency of financial markets...), which, upon examination, prove to be, at least, very fragile. More on this in the next section.)

Another shortcoming of dice is that it relies entirely on the built-in assumption that a single equilibrium exists, which acts as a global attractor no matter how big of a shock the economy goes through. Keen (2021), for instance, shows that, even after the destruction of 90% of the world gdp induced by climatic disaster, the global economy as modeled by dice would still recover and quietly end up in a balanced growth path. This is particularly worrisome when combined with the absence of tail events or sudden consequential climate events on the environmental side of an iam. The latter feature is seen as a substantial limitation by some critics (Pindyck, 2013; Stern et al., 2022). As Stern et al. (2022) put it: 'potentially catastrophic risks are in large measure assumed away by iam.'

Even if the environmental side of conventional iams were more realistic, it would hardly succeed in most cases in permanently dislodging the model from the economic equilibrium on which it was entirely built. Pindyck (2013) shows that this critique extends to a plethora of iams, making them misleading as they give an illusion of knowledge and precision, while constructed on ad-hoc hypotheses, and arbitrary inputs with high sensibility. In our view, the exclusive reliance on a static equilibrium concept is mainly responsible for the inability of conventional iams to cope with the broad lessons of climate science. Indeed, like virtually 'all' sciences since at least half a century, climate science builds on the 'out-of-equilibrium' behavior of 'non linear' dynamical systems. Such an epistemic standpoint is incompatible with that of conventional macroeconomics—probably the sole academic discipline today that clings to a static, equilibrium-based viewpoint.

As for the price of carbon, (Nordhaus 2017) identifies an optimal scc as low as $ 31.2/tCO2-eq by 2030. This value is much lower than the numbers advocated by many experts (see, e.g. (Stiglitz et al. 2017)). Moreover, several authors argue that a carbon tax is a necessary tool but far from sufficient to achieve climate change mitigation goals. They suggest using a wider range of instruments, such as mitigation through reforestation and natural resource protection legislation (Stiglitz et al., 2017). The exclusive focus on the scc and its gross underestimate illustrate how macroeconomics has historically marginalized climate change, reducing it to a simplistic market imperfection problem, overlooking the seriousness and heterogeneity of risks across regions and sectors. We see this epistemic situation as a symptom that, up to recently, an important part of macroeconomics did not take the consequences of global warming seriously.(Some economists reply, on the contrary, that the ipcc, for example, fails to take 'economic science' (i.e. conventional iams) seriously (Murphy 2018). Targets

set by ipcc are viewed as 'so expensive [that] they are worse than doing nothing at all.' Suppose your house is burning, and you are told how expensive it will be to put out the fire and rebuild if you do not act immediately and boldly. If an economist challenges this and recommends fighting with moderation against the flames, should you listen to them and pray that s/he will turn out right or act boldly and immediately? A basic precautionary principle recommends the second attitude, especially since the accumulation of natural disasters in recent years no longer leaves much doubt about the reliability of the prediction of conventional macroeconomics.)

## Damage functions

Exploring economy-climate scenarios raises a fundamental question about our ability to correctly measure the impact of climate change besides, say, the frequency and severity of peaks of heat. Indeed, another key and contentious set of assumptions when describing the interaction of economics and climate deals with the 'damage function'. Introduced as early as in the 70s', it aims at summarizing the impact of the rise in temperatures on gdp and capital. These damages can occur through many channels, including biodiversity loss, extreme weather, natural disasters, loss in agricultural yields and rise of the sea level. Nordhaus (2014) uses a damage function of the form $D = \frac{d^t}{1+d^t}$, where

$$d^t := \psi_1 T + \psi_2 T^2 \quad (3)$$

The choice of a second-order polynomial development for d is far from innocuous: Given a parameterization on minor temperature variations, the parametric shape will yield the smallest effect at high temperatures, leading to our underestimating climate change's impact in 'hot scenarios.' Thus, one may wonder: Why should the Taylor expansion of $d^t$ be of order 2? Nordhaus and Boyer (2000) claim that this choice is based on experts' opinions, which puts the modeling construction at risk of becoming a black box.

Another well-known damage function was introduced by Dietz and Stern (2015), where temperature rise impacts both capital and output:

$$D = 1 - \frac{1}{1 + \sum_{k=1}^{3} \pi_k T^k} \quad \text{Damage level}$$

$$D^k = f_k D \quad \text{Damage on capital}$$

$$D^Y = 1 - \frac{1 - D}{1 - D^k} \quad \text{Damage on production}$$

The presence of a third power, $T^3$, will lead to more pessimistic results than with (3). Figure 1 exhibits three damage functions that provide radically different outcomes for +6°C of global warming relative to preindustrial levels: while (Nordhaus 2014) estimates that less than 10% of global gdp would be lost, (Dietz and Stern 2015) find 90% of gdp loss for the same level of warming. The damage function introduced by the late Weitzman stands somewhere near (Weitzman 2011). Unfortunately, these three functions are almost indistinguishable below a +2°C temperature anomaly. Yet, we cannot afford to wait until that threshold is reached to take action based on which damage function will turn out to be more accurate.

Note that other damage functions have been introduced based on statistical ergodic reasoning (Burke et al. 2015), (Newel 2018). While they are less opaque than the previous functions just
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Climate damage as a percentage of real gdp, as estimated by Nordhaus, (2013), Dietz and Stern (2015), and Weitzman (2010). Source: Bovari et al., 2018a.</figcaption></figure>

alluded to, the fact that they happen to be symmetric concerning the sign of temperature change has led Woillez et al. (2020) to analyze what a cooling of –4°C from preindustrial levels would imply for gdp growth. They end up with surprising, if not utterly implausible, results: some tropical regions would witness an increase of gdp by more than 300%, which would compensate for the loss in GDP inflicted on northern countries, half of which would be buried under several tens of meters of ice, at the end of the last Glacial period, 20 000 years ago. This casts some doubt on the realism of such a damage function and leaves open a broad area of research: how to define damage functions that are estimated in a transparent and reproducible way based on scientific arguments? One could argue that this is an impossible task because, beyond +2°C, the planet will enter a terra incognita for which even paleo-climatologists lack reliable data from the past to assess how our near future might look like. While this a valid objection that should prompt us to be all the more cautious about prospective scenarios, we contend that this should not refrain the scientific community from.

* organizing an educated conversation about damage functions, as long as their construction remains transparent (as in (Burke et al. 2015), (Newel 2018), and (Woillez et al. 2020));

* scrutinizing alternative measures of the impact of global warming, such as the frequency of peaks of heat-humidity (as in (Martin et al. 2023)), that do not depend on the choice of s specific damage function.

* exploring to which extent our future only marginally depends upon the nitty-gritty of damage functions—a conclusion reached by Martin et al. (2023) to which we will turn in the following sections.

## Peaks of heat and humidity

Temperature is not the only cause of economic damages and should not be the only factor, compound climate conditions (Zscheischler et al. 2018) have significant impacts.

Quantifying damage to human health caused by changes in temperature and other key variables, such as humidity, helps assess the impact of global warming on human activities. Mora et al. (2017) have provided a clinically determined curve of combinations of temperature and relative humidity that is lethal to

<figure><img src="image_4.png" /><figcaption>**Figure 2.** (Mora et al. 2017): Mean daily surface air temperature and relative humidity during lethal heat events (crosses) and during periods of equal duration from the same cities but from randomly selected dates (that is, non-lethal heat events; red to yellow gradient indicates the density of such non-lethal events). An SVM (support vector machine) builds a line of lethal conditions. The blue line separates lethal and non-lethal heat events, and the red line is the 95% probability threshold of the SVM; areas to the right of the thresholds are classified as deadly, and those to the left as nondeadly.</figcaption></figure>

average human beings (fig. 2).(Unless they can access air conditioning, average humans die after app. Six hours of exposure to such combinations.) Martin et al. (2023) report a world map with the annual number of days when such lethal conditions would prevail at the end of the century according to a Business-as-Usual scenario as simulated by idee (Fig. 3, panels a,b, c, d. See 6.2 for their model). The average temperature increase would then be around +3.8°C by 2100.

One striking consequence is as follows: when dice recommends a 'socially optimal' path with a maximal climate anomaly of +3.8°C, it indirectly pleas in favor of a world where almost the entire Indian subcontinent, all of East Asia, and much of Central America become uninhabitable by the end of the 21st century. This would represent between 1 and 2 billion climate refugees and, possibly, a repeat of the disaster that struck several colonies at the end of the 19th century following particularly severe El Niño (El Niño Southern Oscillation, enso) phenomena. According to Davis (2005), about 50 million people died from droughts, floods and pandemics. (India, China, Brazil, North Africa, Southern Africa and the Philippines experienced three exceptional climatic periods of drought in 1876–1879, 1889–1891, 1896–1902, during which famine and diseases developed and were responsible for millions of deaths worldwide. Estimates compiled by Davis put the number of deaths at between 31.7 and 61.3 million.)

On the other hand, it is noteworthy that, using a completely different climate model (with no economic back loop), Mora et al. (2017) reach a map of lethal combinations of heat-humidity quite similar to Martin et al. (2023) under the ba u scenario. (Except in the Congo basin, where Martin et al. (2023) seem slightly more optimistic.) The added value of idee, however, is not just to be able to reproduce the ba u path by the literature but to help explore alternative trajectories where public policy might help avoid such a disastrous endgame.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Deadly heatwaves under a scenario close to that advised by dice. a,b represent the number of days spent above the lethal threshold in the years 2068 and 2134, which correspond to the beginning of the irreversible decline phases when integrating the Stern damage function and the Nordhaus damage function into Martin et al. (2023), respectively (see 6.2).</figcaption></figure>

## On the microeconomic foundations of macroeconomics

More often than not, the search for the microeconomic foundations for macroeconomics is considered a must by conventional macro-economists. In our eyes, this leads to two unfortunate consequences, which suggest that this quest is, at the very least, problematic.

First, as should be well known from get, there are inevitable *emergence* phenomena in economics: more is different. This is patent on the consumption side, given the famous Sonnenschein-Mantel-Debreu theorem from 1974 (Mantel 1974), (Debreu 1974): Every continuous and inward-pointing vector field on the non-negative part of the unit sphere of normalized prices can be viewed as the aggregate excess demand function of a well-chosen Arrow-Debreu economy. A much less known result is that the same ‘anything goes’ type of conclusion holds on the supply side. This means, in particular, that the aggregate demand (resp. supply) curves need not by downward (resp. upward) sloping. They may pretty much take the shape of any polynomial so that the set of equilibrium prices may be downright pathological. This negative result has been extended to incomplete markets by Bottazzi and Hens (1996) and to a more restrictive setting where income distribution of the underlying economy is imposed a priori: even with this additional constraint, Chiappori and Ekeland (1999) have shown again that, still, ‘anything goes.’ Following this strand of literature, by presupposing that no emergence phenomenon will ever occur at the macroeconomic level, the search for micro-foundations is at odds with the very theory that it aspires to make its foundation.

Second, most microfounded macroeconomic models also assume perfect foresight of (at least a fraction of) economic households/firms, modeled as maximizing a discounted sum of expected utilities/profits conditional on their current information. (Recently, a strand of literature has tried to consider a hybrid situation where part of the households are Ricardian (in the sense that they share perfect foresight so that the alleged ‘Ricardian equivalence’ of public spending holds) while part of them are not. While this is undoubtedly a step towards more realistic modeling of households, the non-negligible fraction of ‘perfectly rational’ agents remains key for this type of analysis.)

A major stumbling block on this theoretical trail was discovered several decades ago by (Cass and Stiglitz 1970). In today's parlance, their finding can be formulated as follows: 'rational expectations' are not eductively stable in the sense of (Evans et al. 2019). In other words, even if individual agents were perfectly 'rational' (and thus, endowed with an unbounded memory and computing power), the optimal aggregate trajectory followed by an economy populated by such agents would not be stable to the slightest deviation from 'rational expectations': as soon as a single agent, at one point of time, infinitesimally deviates from perfectly anticipating the future conditionally on the past information she has access to, the resulting trajectory of the economy will never come back on a course compatible with such 'rational expectations.' Otherwise stated, in the 'enlarged phase space' of an economy obtained by stacking the space of expectations onto that of the usual state variables, solutions compliant with 'rational expectations' are locally unstable. Imperfect foresight is the rule, and 'rational expectations' are as implausible as the locally unstable steady state of the oscillating pendulum in the high school exercise of classical mechanics.

Despite the relative neglect of these two types of impossibility results—the inadequacy of the ‘representative agent’ trick and the lack of eductive stability of ‘rational expectations’—by conventional climate macroeconomics, we should take these pretty serious setbacks into account. Bounded rationality and emergence are unavoidable in macroeconomics. In other words, it is time to cope with complexity.

## Complexity economics

The growing literature on social-ecological systems now widely recognizes that such systems can be characterized as intertwined, complex adaptive systems (Folke et al. 2003; Norberg and Cumming 2008; Schoon and Van Der Leeuw 2015; Preiser et al. 2018; De Vos et al. 2019; Schlüter et al. 2019). This implies, among other things, that social-ecological organizations need to be understood as context-sensitive and adaptive systems, which emerge as the outcome of multiple nonlinear, recursive, causal interactions causing spill-over effects across different spatial and temporal scales (Levin and Hesshaimer 2000).
Complexity economics burgeoned in the early 2000s (Elsner 2004; Beinhocker 2006; Arthur 2021). It aims at fixing the misrepresentation of dynamics in conventional, marginalist economics. Instead of assuming that economic agents will automatically and instantaneously coordinate on a (Nash, Walrasian ...) equilibrium, it tries to qualify under which conditions they may or may not reach equilibrium along what path. To our knowledge, this methodological endeavor has been mostly applied to microeconomics.(There are, of course, a few exceptions; see (Wells 2012) and (Fang and Sanglier 1997) for instance.) The following subsections suggest a direction of research that might be viewed as a contribution to complex macroeconomics.

## On the importance of time

An uncomfortable but promising middle ground between the traditional, static equilibrium approach and the kind of dynamics we seek consists of viewing economic kinetics as a succession of short-run equilibria, as in the 'temporary equilibrium' literature (Grandmont, 1977). With a different formalism, it was continued by Champsaur and Cornet 1990; Giraud and Tsomocos 2010). With little follow-up, however. There are good reasons for that: learning to walk upright is definitely not a matter of collecting a series of successive balancing positions. Still, risking a position of imbalance may lead to a new equilibrium state—one step further. Many economists of different backgrounds have long advocated this point. Joan Robinson phrases a similar idea in Robinson (1974) in her call for modeling non-equilibrium dynamics. Samuelson (1966) pleaded in favor of the Austrian conception of capital as time, so that the productivity of capital is the productivity of time itself. But, if this is so, an economic theory of capital must rely on a proper modeling of time and not on its methodological concealment, as induced by the equilibrium postulate in general equilibrium theory (get) (see Debreu, 1959).

To widen the usual macroeconomic focus from equilibrium-based statics to far-from-equilibrium dynamics is challenging in many ways. Let us see how one could proceed in the simplest possible way.

## Stock-flow consistency

One simple criterion that should be satisfied by out-of-equilibrium trajectories is stock-flow consistency (Godley and Lavoie, 2007; Taylor 2021, chapter 1), which states that, at any point in time, the stocks on the balance sheet of every economic entity are the exact outcome of the history of flows this entity came across. This requirement might sound elementary, but it is not.(Especially when embedded into the first two laws of thermodynamics.) To begin with, assessing whether SFC is satisfied in a standard GE model is hard. Indeed, everything is supposed to take place 'at equilibrium', and the distinction between stocks and flows becomes blurry. Consider, for instance, final consumption vectors or initial endowments in an Arrow-Debreu economy: are these flows or stocks? Who can tell? Out-of-equilibrium, stock-flow consistency (SFC) is a must. It does provide some bite, ensuring that, across time, no monetary aggregate, matter or energy is created out of nowhere or disappears.(This does not preclude money creation by banking credit.) As the first rule we should observe, sfc provides the skeleton of any macroeconomic model, in the form of a set of structural and accounting constraints on its variables, on top of which differential equations describing the dynamics of those variables can be laid out.

As an example, consider the Social Accounting Matrix of Table 1, which describes the balance sheet, transactions and

flow of funds in a mono sectoral, closed economy, $\mathcal{E}$, comprising households, private firms and commercial banks.

Table 1 includes inventories of unsold products, V, as a consequence of Say's law way being possibly violated. Output is therefore given by the sum of consumption, C, investment I, and changes in inventories: $Y(t) = C(t) + I(t) + \dot{V}(t)$. $p(t)$ stands for some price deflator, nominal income flows are given by $Y_n(t) := p(t)(C(t) + I(t)) + c(t)\dot{V}(t)$, where c is the unitary production cost: $c(t) := W(t)/Y(t)$, with $W(t) := w(t)L(t)$ the wage bill of the economy, $w(t)$, the unitary wage and $L(t)$, employed labor force.

Households' assets consist of firms' equity, $E_f(t)$, bank equity, $E_b(t)$, and current money deposits, $M_h(t)$. Their liabilities consist of loans, $L_h(t)$, resulting in net worth (i.e. wealth) $X_h(t) := E(t) + M_h(t) - L_h(t)$, with $E(t) := E_f(t) + E_b(t)$. Firms' assets are capital stock, $p(t)K(t)$, plus inventories, $c(t)V(t)$ (where c(t) stands for the unitary production cost), and current money deposits, $M_f(t)$. In contrast, their liabilities are their loans, $L_f(t)$, yielding equity $E_f(t) := p(t)K(t) + c(t)V(t) + M_f(t) - L_f(t)$(Here, equity is treated as a balancing variable, so that the net worth of non-financial firms is always $X_f(t) = 0$.). Similarly, loans, $L(t) := L_h(t) + L_f(t)$ are the assets of the banking sector. Its liabilities are deposits, $M(t) := M_h(t) + M_f(t)$, whereas bank equity is $E_b(t) = L(t) - M(t)$. Hence, the net worth of firms and banks remains $X_f(t) = X_h(t) = 0$. For simplicity, we assume that firms do not keep positive deposits, preferring to use any balances to repay their loans instead. Retained earnings are reinvested:

$$ \Pi_f(t) - \Delta_f(t) \equiv S_f = I_f, \quad (4) $$

where $\Delta_f \ge 0$ designates dividends distributed by the corporate sector to their shareholders. Equality of investment and saving in (4) has been the source of much confusion about the origin of Money (see (Werner 2016) and the references therein) that cannot be discussed at length in this paper. Suffice it to say that the unique way to make sense of banking credit is to admit that causality runs from I to S (contrary to the traditional wisdom). It is the gist of *emerging phenomena* in macroeconomics. Indeed, as observed (e.g. by (Taylor 2021)), savings and investment flows must be balanced *independently* of the (free) will of economic actors. Already in *The General Theory*, John Maynard Keynes noticed that:

> "The reconciliation of the identity between saving and investment with the individual's apparent 'free will' to save irrespective of what he or others may be investing essentially depends on saving being [...] a two-sided affair. For [...] the reactions of the amount of his consumption on the incomes of others makes it impossible for all individuals simultaneously to save any given sums. Every [...] attempt to save more by reducing consumption will so affect incomes that the attempt necessarily defeats itself" (Keynes 1936).

Stock-flow consistency being the backbone, let us provide flesh to our model by adding metabolic rules (production, degradation, transformations on the physical variables) and behavioral rules (investment, inflation, wage valuation ...). This will allow studying propagations of any disturbances in the system and the associated feedback loops (or homeostatic feedbacks) that might amplify or dampen any fluctuations from equilibrium.

The first rule, namely capital accumulation, can hardly be dispensed with:

$$ \dot{K} = I - \delta K. \quad (5) $$

This is the driving equation of the Perpetual Inventory Method used by accountants to build capital time series. In reality, there
<table><caption>**Table 1.** Social Accounting Matrix: balance sheet, transactions, and flow of funds for the closed economy with inventories and prices</caption><thead><tr><th></th><th>Households</th><th>Firms</th><th>Banks</th><th>Row Sum</th></tr></thead><tbody><tr><td><strong>Balance sheet</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Capital stock</td><td></td><td>+pK</td><td></td><td>pK</td></tr><tr><td>Inventory</td><td></td><td>+cV</td><td></td><td>cV</td></tr><tr><td>Deposits</td><td>+M<sub>h</sub></td><td>+M<sub>f</sub></td><td>-M</td><td>0</td></tr><tr><td>Loans</td><td>-L<sub>h</sub></td><td>-L<sub>f</sub></td><td>L</td><td>0</td></tr><tr><td>Equity</td><td>E</td><td>-E<sub>f</sub></td><td>-E<sub>b</sub></td><td>0</td></tr><tr><td>Column sum (net worth)</td><td>X<sub>h</sub></td><td>X<sub>f</sub> = 0</td><td>X<sub>b</sub> = 0</td><td>X</td></tr><tr><td><strong>Transactions</strong></td><td></td><td>Current</td><td>Capital</td><td></td></tr><tr><td>Consumption</td><td>-pC</td><td>+pC</td><td></td><td>0</td></tr><tr><td>Investment</td><td></td><td>+pI</td><td>-pI</td><td>0</td></tr><tr><td>Change in Inventory</td><td></td><td>+cV̇</td><td>-cV̇</td><td>0</td></tr><tr><td>Depreciation</td><td></td><td>-pδK</td><td>+pδK</td><td>0</td></tr><tr><td>Wages</td><td>+wL</td><td>-wL</td><td></td><td>0</td></tr><tr><td>Interest on loans</td><td>-rL<sub>h</sub></td><td>-rL<sub>f</sub></td><td>+rL</td><td>0</td></tr><tr><td>Interest on deposits</td><td>+rM<sub>h</sub></td><td>+rM<sub>f</sub></td><td>-rM</td><td>0</td></tr><tr><td>Pre-dividend net profits</td><td></td><td>-Π<sub>f</sub></td><td>+Π<sub>f</sub></td><td>0</td></tr><tr><td>Dividends</td><td>Δ<sub>f</sub> + Δ<sub>b</sub></td><td></td><td>-Δ<sub>f</sub></td><td>0</td></tr><tr><td>Column sum (balances)</td><td>S<sub>h</sub></td><td>0</td><td>S<sub>f</sub> - p(I - δK) - cV̇</td><td>0</td></tr><tr><td><strong>Flows of funds</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Change in capital stock</td><td></td><td>pK̇</td><td></td><td>pK̇</td></tr><tr><td>Change in inventory</td><td></td><td>cV̇</td><td></td><td>cV̇</td></tr><tr><td>Change in deposits</td><td>+Ṁ<sub>h</sub></td><td>+Ṁ<sub>f</sub></td><td>-Ṁ</td><td>0</td></tr><tr><td>Change in loans</td><td>-L̇<sub>h</sub></td><td>-L̇<sub>f</sub></td><td>+L̇</td><td>0</td></tr><tr><td>Column sum (savings)</td><td>S<sub>h</sub></td><td>S<sub>f</sub></td><td>S<sub>b</sub></td><td>pK̇ + cV̇</td></tr><tr><td>Change in firm equity</td><td>+Ė<sub>f</sub></td><td>- (S<sub>f</sub> + pK̇ + cV̇)</td><td></td><td>0</td></tr><tr><td>Change in bank equity</td><td>+Ė<sub>b</sub></td><td></td><td>-S<sub>b</sub></td><td>0</td></tr><tr><td>Change in net worth</td><td>Ė + S<sub>h</sub></td><td></td><td>(pK̇) + (cV̇)</td><td></td></tr></tbody></table>

is an apparent delay between investment and the production of capital—as was forcefully underlined by Kalecki (1968). Adding a delay, however, quickly leads to chaotic behavior (Matsumoto and Szidarovszky 2020), and we shall refrain from doing it in this paper.

Besides capital accumulation, we need several other dynamical equations to flesh out our models. The next section discusses models inspired by the Lotka–Voltera dynamics inherited from evolutionary biology.

## Lotka–Voltera models in macroeconomics
### The short run Phillips curve

Goodwin (1967), Akerlof and Stiglitz (1969), and Van der Ploeg (1985) further specify the system behavior that compares Lotka–Voltera dynamics: employment is the prey while the wage share (i.e. the fraction of total incomes accrued to workers as labor compensations) is the predator. The rationale behind their partially antagonistic interplay goes as follows. As activity increases, hires accelerate; the growing strain on the labor market makes it easier for workers to negotiate an increase in money wages. Thus, wages should increase more rapidly so that the wage share ($\omega := wLpY \in [0, 1]$) will rise. Consequently, the profit share ($\pi := 1 - \omega$) must dwindle. As investment becomes less profitable, private firms will adjust their investment strategy downward. Hence, growth will slow down. The employment rate should decrease, making it harder for employees to get a pay rise. Consequently, the wage share ($\omega$) should drop back. Up to the point where profitability is restored, investments will accelerate again—the economy then enters a new business cycle.

This simple but appealing narrative relies on a relationship between the employment rate, $\lambda := LN \in [0, 1]$(where $N > 0$ is the

size of the working population.) and wages, known as the short run Phillips curve (Phillips 1958; Samuelson and Solow 1960). It can be formalized by assuming that there exists some continuous, increasing real-valued function, $\phi$, such that:

$$ \frac{\dot{\omega}}{\omega} = \phi(\lambda) \qquad (6) $$

As is well known, the Phillips curve is somewhat controversial. On the one hand, it has long been a workhorse model of wage–employment relationships. Still, today, the correlation between employment and wage growth is empirically patent (see Fig. 4). On the other hand, and despite Phillips’ original paper—(Phillips 1958), having been erroneously interpreted as a relationship between employment and prices—the ‘Phillips curve’ is sometimes declared dead or, less dramatically, blamed for having failed to predict both the stable inflation seen in the aftermath of the global financial crisis in the 2007–9 period—the ‘missing deflation’ puzzle—and the stable inflation during the following recovery period—the ‘missing inflation’ puzzle(Such properties are easier to access through multi-sectoral dynamics, regionalization and the explicit introduction of money). Nonetheless, Phillips’ seminal idea underlying the wage–employment relationship is still relevant (Gordon, 2013; Ball et al., 2022) if one grants that (i) the channel between wages and prices is not a trivial one and might have played a role in the two puzzles just alluded to; (ii) institutional changes can deform the graph of $\phi$, different methodologies of employment measurement, as well as shifts in the bargaining power of, say, trade-unions (Nalewaik, 2016; Ratner and Sim, 2022).
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Employment rate and hourly earnings, US, 1988–2019, fed. Res. Econ. Data, St Louis Fed. Blue: employment; green: wage percentage change.</figcaption></figure>

One way to account for these phenomena and the variety of institutional changes that affect an economy's social fabric—namely day-to-day wage bargaining—is to allow context-dependent structural breaks to affect the shape of $\phi$ across time. McIsaac (2021), for instance, provides convincing empirical evidence that the us labor market has experienced two structural breaks since the postwar period. This dependency forces us to relinquish the quest for ergodic economic dynamics and abstract, general economic ‘laws’. Instead, we must enter into the nitty-gritty of a country's or a region's institutional and historical specifics. This is, after all, a hallmark of complex systems, which we claim social-ecological systems are.

## Lotka–Volterra models of employment and wages

The predator–prey dynamics first introduced by Goodwin (1967) relies on the following additional assumptions:

* the supply side is a two-factor Leontief technology Y := min{$Kv; aL$}, with labor adjusting to maximal production so that

$$Y = \frac{K}{v} = aL, \qquad (7)$$

where $a > 0$ measures Harrod-neutral, labor-enhancing technological progress and $v > 0$ is the (constant) capital-output ratio.

* profits, $\Pi := pY - wL$, are entirely reinvested, so that investment is $I := \Pi$;

* wages are entirely spent into consumption $C = wL$ (Using these investment and consumption behavioral hypotheses, Say's law is automatically fulfilled.).

The resulting dynamics can be summarized by the following dynamical system, with $\omega = wLpY$ the wage share:

$$\begin{cases} \dot{\omega} = \omega(\phi(\lambda) - \alpha) \\ \dot{\lambda} = \lambda\left(\frac{1-\omega}{v} - [\alpha + n + \delta]\right) \end{cases} \qquad (8)$$

Here, $\alpha = \dot{a}/a > 0$, $n > 0$ is the growth rate of labor forces. The first equation in (8) means that the wage share increases as long as wages grow more rapidly than labor productivity. The second says that the employment rate will rise if the growth rate of output

$$g := \frac{\dot{Y}}{Y} = \frac{1-\omega}{v} - \delta = \frac{\Pi}{pK}$$

remains higher than the sum of labor productivity and labor force growth rates.

Fig. 5 provides a set of trajectories of the Goodwin model for different initial values, in the case of a Phillips curve verifying

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Set of trajectories for a Goodwin model around the Solovian equilibrium, each with the same parameter values and different initial conditions.</figcaption></figure>

$\lim_{\lambda \to 1^-} \phi(\lambda) = \infty$. Solution trajectories are all closed orbits with cycles occurring at a specific frequency. The dynamics are conservative and admit a unique steady state:

$$(\omega_{eq}, \lambda_{eq}) := (1 - v(\delta + \alpha + n), \phi^{-1}(\alpha))$$

For a small deviation from this equilibrium, the frequency of the business cycle will be:

$$f_0 = \sqrt{\frac{\lambda_{eq} \omega_{eq} \phi'(\lambda_{eq})}{v}}$$

The model reproduces the endogenous emergence of real business cycles but does not account for crises.

## Factor substitutability: The Van der Ploeg model

(Van der Ploeg 1985) generalizes the Goodwin model by allowing for partially substitutable production factors. For that purpose, the Leontief structure is replaced by a two-factor CES production function with $\zeta = \frac{1}{1+\eta}$ as the short run elasticity between capital and labor. This comes at the cost of strengthening the (implicit) assumption of profit maximization (or cost minimization) encapsulated in (7): Now, one assumes that firms instantaneously choose L to equate the marginal productivity of labor with
real wages. This leads to an endogenously time-varying capital-output ratio:

$$v(t) := \frac{K(t)}{Y(t)} = \frac{1}{A} \left( \frac{1 - \omega(t)}{b} \right)^{-\frac{1}{\eta}},$$

where $b \in [0, 1]$ stands for the capital intensity of production and $A > 0$ is total factor productivity. Switching from a Leontief to a ces production function turns the dynamical system into a dissipative one: When workers manage to obtain a large share of total income via wages, firms find it less profitable to hire workers and therefore switch away from labor to machinery. Consequently, solution trajectories still exhibit cycles, but now, oscillations are damped. In the long run, the economy converges towards some globally asymptotically stable long-run steady state, namely the analog of the unique equilibrium of the Solow model.

In our eyes, the most interesting version of the production side within this family of prey-predatory models has been introduced by (Akerlof and Stiglitz 1969) using a Putty-Clay capital structure. In a sense, it provides the right intermediary solution putting both the Leontief and the ces versions on equal feet. However, we will keep the Putty-Putty or the Clay-Clay structure in the rest of our discussion to go directly to the essentials.

## Corporate debt: The Keen model

Refining this basic prey-predator setting is now possible by adding other vector fields that capture additional stylized facts beyond the interplay between wages and underemployment. Both dynamical models discussed above assume investment to be entirely and exclusively financed through current profits. This ignores the role played by the banking sector in financing investment and the possibility for households to accumulate savings. (Keen 1995) amended the Goodwin model by defining investment as $I := \kappa(\pi)Y$, with $\kappa$, a continuously increasing function. Investment is now allowed to be larger (or lower) than profits:

$$\dot{D} = I - \Pi = pY(\kappa(\pi) - \pi) \quad (9)$$

with $D$ standing for aggregate corporate debt and profits being defined net of debt servicing: $\Pi = Y - W - rD$, where $r \ge 0$ is the (constant, for now) short run interest rate. Denoting $\dot{d} := DY$ the debt-to-output ratio, and taking a ces production function, adding (9) yields the following 3-dimensional dynamical system:

$$\left\{
\begin{aligned}
\dot{\omega} &= \omega \left( \frac{\eta}{1+\eta} (\phi(\lambda) - \alpha) \right) \\
\dot{\lambda} &= \lambda \left( \kappa(\pi) A \left( \frac{1-\omega}{b} \right)^{\frac{1}{\eta}} - \delta - \alpha - n - \frac{1}{\eta(1-\omega)} \frac{\dot{\omega}}{\omega} \right) \\
\dot{d} &= d \left( r - \kappa(\pi) A \left( \frac{1-\omega}{b} \right)^{\frac{1}{\eta}} + \delta + \frac{\dot{\omega}}{\eta(1-\omega)} \right) + \kappa(\pi) - (1-\omega)
\end{aligned}
\right.
\quad (10)$$

with the profit share, $\pi = 1 - \omega - rd$, as an auxiliary variable.

Analytically, this family of models shares a set of interesting properties beyond generating endogenous real business cycles.

First, they display multiple equilibria—a feature that is hard to get in a general equilibrium framework and nevertheless is viewed as a must (see, e.g. (Vines and Wills 2020)). Among these, two types of equilibria show up as being most interesting—both because they are locally asymptotically stable under reasonable conditions and economically meaningful. The first one is an interior steady state, analogous to the Solovian equilibrium and sometimes called the ‘good equilibrium,’ where the economy stabilizes in the long run at a finite debt ratio, positive wage share, and positive employment rate. Generally, this equilibrium satisfies the

Golden rule balanced growth property that was the hallmark of the (unique and globally stable) fixed point of the Solow model: in the long run, output and capital grow simultaneously, given by $\alpha+n$. But, at variance with the Solow set-up, there is also a second steady state, namely, a catastrophic equilibrium displaying an infinite debt ratio and vanishing production and wages. Private debt accumulates too fast around this Fisherian debt-deflationary equilibrium, so the productive sector sinks into an unescapable debt overhang. Figure 6 overviews the diversity of trajectories around these equilibria. The stability properties of each equilibrium, the geometry of their basin of attraction, and the speed at which an economy converges to each are all model-specific. For instance, and somewhat unsurprisingly, the Solovian equilibrium basin of attraction in the Van der Ploeg model is usually larger than in the case of complementary factors of production. At the same time, the amplitude of cycles is generally smaller.

At variance with static equilibrium approaches, these models enable us to understand how the state of an economy can wander from one equilibrium to another within the phase space, a crucial feature for studying the impact of climate change.

## Climate damage and economic destabilization
### Coping with collapse

Motivated by the failures of traditional iams and the Stern-Stiglitz criticism, some authors rely on complexity economics to study climate-economy models. In particular, (Bovari et al. 2018a) introduced a dynamics calibrated at the world scale that aims at modeling the possibility of extreme climate risks and financial instability and the interaction between those risks. They couple an elementary stock-flow consistent dynamics inspired by (Keen 1995) with the simplified three-layer climate back-loop used in Nordhaus’ dice 2016R model. The two modules are linked through a damage function and carbon pricing instruments, allowing the authors to identify two sources of global instability: global warming and the over-indebtedness of the private sector. In this way, the risks inherent to transitioning to a low-carbon world economy that Carney alluded to can be formally studied.

Let $\pi := \frac{\eta}{pY}$ be the profit share. (Bovari et al. 2018a) stipulate an investment share function, $\kappa(\pi)$, and a dividend share, $\Delta(\pi)$: the former quantifies the fraction of real output dedicated to investment; the latter, the fraction of nominal output distributed as dividends to shareholders. The depreciation of capital $\delta_D$ is the sum of a ‘natural’ depreciation, $\delta$, and a climate-induced destruction rate, $D^K$:

$$\left\{
\begin{array}{l}
I = \kappa(\pi)Y \\
\dot{K} = I - \delta_D K \\
\dot{D} = pI + \Delta(\pi)pY - \Pi - p\delta_D K
\end{array}
\right.
$$

As for firms’ net profit, it now depends on the magnitude of damage to capital.

$$\Pi := pY - W - rD - pT_f - p\delta_D K$$

where Y stands for the final output net of climate damage. The behavior of the production sector thus incorporates the impact of global warming since investment and dividend decisions depend upon net profit. Moreover, imperfect competition with constant markups $\mu$ is assumed, together with cost-pushed inflation, with
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Typical trajectories generated by Goodwin dynamics, with various initial conditions. The left and central cases correspond to an initial debt ratio of resp. $d(t_0) = 0$ and $d(t_0) = 500\%$; both eventually converge towards the 'good' equilibrium. The right-hand case starts with a larger initial debt ratio, $d(t_0) = 800\%$, that stifles the economy and leads to a debt crisis.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Economy trajectories $(\omega, \lambda)$, as generated by the Goodwin-Keen model with damage functions. Left: With Nordhaus' optimistic damage function. Right: With Dietz-Stern's catastrophic damage function.</figcaption></figure>

relaxation time $\frac{1}{\eta_p}$:

$$i := \frac{\dot{p}}{p} = \eta_p (\mu cp - 1).$$

Since the unitary cost, c, includes the impact of warming on gdp, this price dynamics presupposes that firms pass on the part of the cost of emitting co₂-e to their customers. (The article also assumes a sigmoid population growth: $\frac{N}{N_0} = n(N) = n_0 (1 - NN_\infty)$ based on the un median demographic scenario.)

The model couples economy and climate with two elements: a damage function à la Nordhaus-Sztorc and a parametric carbon price instrument designed to avoid the difficulty of choosing a discount rate. The impact of climate change is then studied through four scenarios, with varying assumptions about the damage function.

Absent climate change, the world economy would converge slowly towards a good equilibrium, becoming stationary around the 23rd century. When climate change enters the picture, it affects the economy in both qualitatively and quantitatively different ways, depending on which damage function is chosen. Figure 7 shows the various trajectories of two of the three critical variables in Goodwin-Keen: wage share and employment rate:

* With the optimistic damage function from (Nordhaus 2014), climate change does tend to drive the phase state, $(\lambda, \omega, d)$, away

from the good long-run steady state, yet market forces act as stabilizing forces. After a transitory period, the economy locks onto a stabilizing trajectory. In the end, the economy still converges to the Solovian equilibrium.

* However, with the catastrophic damage function from Dietz and Stern (2015), the severity of destruction inflicted by climate change overwhelms market forces so that the whole economy undergoes a path of 'degrowth by disaster' (not by design), and eventually collapses, while the employment rate and wage shares fall to zero.

The model allows us to understand better the interaction between economic, financial and climate instability. On policy recommendations, authors proposed carbon price trajectories compatible with achieving a +2.5°C maximal temperature anomaly, with a minimum required trajectory of US$44 per tCO₂-e in 2020, $140 in 2030 and $300 in 2040. Figure 7 also shows how the Solovian steady state itself is affected by climate change, which is less desirable as the planet gets warmer. In particular, with the Stern damage function, the debt ratio in the Solovian equilibrium can reach 3 for 2°C of warming and about 5 for 3°C of warming. The wage share also shrinks significantly, so inequalities widen with climate change, even in a good equilibrium. Of course, some essential phenomena are uncertain, such as labor productivity and some climate resilience variables. When we have information about the first moments of the distribution of these variables, we
can use Monte-Carlo simulations to generate a range of likely scenarios, such as what was done by Stiglitz et al. (2017)) or by Bovari et al. (2019). In these reports, the business-as-usual scenario looks ugly, with a steady degrowth in actual output and exploding deflation before the end of the century. Although there is a lot of uncertainty involved, these results are significant.

Such dynamic approaches can be used to answer questions, such as: In which basin of attraction are we currently locked? Given estimates from climate science, which policies will ensure that we converge towards a good equilibrium? Among such policies and trajectories, which are more desirable?

## Climate models and IDEEs

### Towards better climate modules: A simple taxonomy of climate models

Climate models vary in spatial resolution, coverage and complexity in rendering the physical processes they aim to simulate. They are typically validated through hind-casting tests before being trusted to produce prospective scenarios. Such scenarios are used to produce the reports of the Intergovernmental Panel for Climate Change (ipcc, see e.g. (Nakicenovic et al. 2000).

General Circulation Models (gcms) are the most realistic—but computationally heaviest—models. They have global coverage and aim to model multiple atmospheric and oceanic layers and the exchange of matter and energy between them. Although gcms aim to produce the most accurate, fine-resolution results based on detailed, high-dimensional models, they are typically computationally intensive. Some require weeks to months of runtime to produce global scenarios. This makes them of limited practical use for coupled climate-economy models. Ideally, we seek low-dimensional models that can reproduce some physical processes satisfyingly, i.e. approximate laws from fluid dynamics and thermodynamics, while running much faster. This is the motivation behind Earth Models of Intermediate Complexity (emics, Claussen et al. 2002), which aim to compromise as little as possible on the physical processes they model while drastically reducing the dimension of the system, both by limiting their spatial and/or temporal resolution, and by considering fewer variables, either through statistical methods or by design. With Climber, Saltzman (1978) and Petoukhov et al. (2000) achieve this by trading fluid dynamic processes with a low-dimensional statistical process. At the same time, Quasi-Geostrophic Models are built on a reduced form of the nonlinear Partial Differential Equations used to approximate global atmospheric dynamics. One such model is i LoveClim (Goosse et al., 2010; Roche, 2013; Roche et al., 2014; Bügelmayer et al., 2015; Quiquet et al. 2018). It is based on a three-layer model of the atmosphere where the equation of the Quasi-Geostrophicity of the Potential Vorticity is solved, together with an oceanic model (which includes an advection equation for salinity) and a vegetation module.

## A more realistic economy-environment coupling: IDEE

Martin et al. (2023) extend and expand the work of Bovari et al. (2018a), Bovari et al. (2018b), and Bovari et al. (2019) by replacing the elementary climate module from dice with i LoveClim, enabling a much richer dynamical coupling of a spatial biosphere together with the economics sphere: emissions from the economic model are used as input at each timestep of the climate model, giving a distribution of the temperature on the globe, sent back on the economy through an aggregated damage function. The resulting model coined an Integrated Dynamic Environment-Economic (idee) allows the capture of the interplay of out-of-

equilibrium dynamics of wages, employment, debt, inflation and so on, with atmospheric, oceanic and vegetation variables over the entire planet.

Crucially, the economic module also introduces endogenous technical progress. Endogenous growth is captured by the 'Kaldor-Verdoorn law' (Kaldor, 1957; Romer 1983), introduced and studied in this type of model by Goodwin (1991) for a Leontiev production function and Aguiar-Conraria (2008) for a CES:

$$ \frac{\dot{a}}{a} = \alpha + \gamma_g g, \qquad (11) $$

where, usually, $\gamma_g \sim 0.5$. As is well known, Eq. (11) causes hysteresis, i.e. path-dependence, in the overall economic dynamics inside a cycle. However, this widens the basin of attraction of the Solovian catastrophic equilibrium. At variance with the exogenous growth case, the balanced growth path becomes locally unstable for a significant range of reasonable parameters. As the 'bad' equilibrium undergoes a Hopf bifurcation, as $\gamma_g > 0$ increases, the catastrophic equilibrium becomes unstable and, instead, a limit-cycle branch from the fixed point. Simulations suggest that the bifurcation usually occurs for $\gamma_g$ lower than 0.5. Therefore, a tentative educated conclusion is that endogenous growth presumably destabilizes both equilibria. Economic instability is probably something we must live with. When coupled with a climate damage function, as in Martin et al. (2023), these dynamics imply that technical progress, or the lack thereof, is endogenously determined by the influence of global warming on investment and real economic growth. In particular, degrowth driven by catastrophic damage induced by too hot a planet can no longer be artificially counteracted by some exogenous technical progress. Degrowth becomes self-reinforcing.

Still, on the economic side, the second innovation of Martin et al. (2023) is introducing default and collateral. Besides being more realistic, this additional feature addresses a controversy about the Minsky financial instability hypothesis (Minsky, 1982). At the Fisherian equilibrium of the basic Goodwin-Keen dynamics, firms keep investing even though they no longer profit. In our view, Pottier and Nguyen-Huu (2017)) rightly challenge the popular interpretation of the debt-crisis equilibrium as the hallmark of Minskyan financial instability by questioning the plausibility of firms' behaviors around the 'bad' equilibrium: why would firms keep investing even though doing so further sinks them into debt? When, instead, firms disinvest in their neighborhood, the Fisherian equilibrium is no longer an attractor, and the economy instead displays a limit cycle.

Martin et al. (2023) and Giraud et al. (2023) address this criticism by constraining debt with collaterals. (In get (Geanakopoulos and Zame 2014), (Fostel and Geanakoplos 2008) have introduced collateral requirements which are the basis of the hedging mechanics against credit risk that unraveled in the 2008 global financial crisis (gfc).) As already noted, current profits may not suffice to finance the whole of i, in which case firms have to borrow from the banking sector. This time, however, as with much lending in modern economies, corporate debt is secured by some collateral, namely the current stock of capital valued at its market price, pK. Collateral is assumed to be held and used by the borrower, and forfeiture of collateral is the only consequence of default. Borrowers will, therefore, always deliver $\min\{D, pK\}$, the minimum of what is promised and the collateral's value. Knowing this, lenders need not worry about the identity of the borrowers but only about the future value of the collateral. Consequently, we can restrict attention to firms' aggregate debt
and assume that, as D grows closer to its upper bound, pK, many companies will go bankrupt, following complex micro-economic patterns and interdependencies whose nitty-gritty is neglected. Rather, as a proxy, the authors posit that the production sector no longer invests whenever D = pK, i.e. $(1 - \delta_Y)d = v$. Indeed, if firms exhibit a nominal debt greater than their collateral, they have no incentive to pay back their debt; as a result, banks will stop lending to them anyway. Moreover, default is accompanied by a transfer of ownership over the collateral from borrowers to lenders. For simplicity, they assume that, for a given debt ratio, d, the proportion.

$$ \Gamma(d) = 1 - \exp\left(-\frac{Ad^2}{v^2 - d^2}\right) \in [0, 1], \quad (12) $$

The banking sector seizes capital and, therefore, no longer operates. For $A > 0$ small enough, Eq. (12) amounts to approximating a Dirac mass at $d = v$. Consequently, the private debt ratio is endogenously bounded from above by the capital-output ratio, $v$.

Again, Giraud et al. (2023) shows that the dynamics with default admits at least the two equilibria we are now familiar with. However, as with endogenous growth, both tend to be unstable for a wide range of reasonable parameters. They provide sufficient conditions under which, nevertheless, there exists a compact attractor that is locally asymptotically stable—either a limit-cycle or some compactum, of which no single point is individually stable. (Notice that this result is not the trivial consequence of the 2-dimensional Bendixson-Poincaré theorem.)

Martin et al. (2023) introduce capital collateral for firms' loans in the same way as Giraud et al. (2023). Considering damages induced by global warming, the debt-to-output ratio is, therefore, bounded from above by $v/\delta_Y$. As for investment, it is capped by the following:

$$ I \le \kappa(\pi)Y \left(1 - \frac{(1 - \delta_Y)d}{v}\right)^{\frac{1}{\lambda}} \quad (13) $$

Surprisingly, both the optimistic Nordhaus damage function and the catastrophic one, introduced by Dietz and Stern, lead to a collapse in actual output, wages, and employment, together with corporate debt stuck to its ceiling, $v$. The main difference between these two extreme damage functions is the time horizon of worldwide degrowth by disaster in the Business-as-Usual (ba u) scenario: it takes place as soon as in 2080 in the Dietz-Stern case, and a little more than half a century later in the Nordhaus case. Consequently, the speculation over the damage function becomes somewhat less decisive as, in both cases, the ba u path is not sustainable. The question is no longer 'if' but 'when' our global economy would lock onto a 'financial Minsky moment' (Carney) induced by global warming. In this perspective, strong abatement is needed to get closer to an efficient long-run equilibrium and, more bluntly, avoid a catastrophic endgame. Further, the authors identify a temperature anomaly threshold between +2.3°C and +3.8°C (depending upon the chosen damage function) as a financial tipping point. Indeed, they show that whatever the (public) policy put in place beyond this threshold, the world's productive sector would inevitably decline and end up trapped in a debt overhang. In addition to environmental and climate risks, this provides another new justification for the Paris Agreement target well below +2°C: financial stability requires the temperature anomaly is not too far from the +2°C limit.

The authors also show that a carbon tax policy helps avoid a catastrophic endgame without jeopardizing public finances. To be

<table><caption>**Figure 8.** Relative average distance to true observations between Goodwin model and var, reported in table 8 of (McIsaac 2021). The lower the score below 1, the higher the overtaking of a var by the Goodwin model. The variable $h$ denotes the number of years ahead considered for prediction.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Leontief</th><th colspan="2">CES</th></tr><tr><th>ω</th><th>λ</th><th>ω</th><th>λ</th></tr></thead><tbody><tr><th>h = 1</th><td>0.13921</td><td>0.10817</td><td>0.13712</td><td>0.10706</td></tr><tr><th>h = 2</th><td>0.16750</td><td>0.19527</td><td>0.17581</td><td>0.18468</td></tr><tr><th>h = 3</th><td>0.19836</td><td>0.29957</td><td>0.21159</td><td>0.27826</td></tr><tr><th>h = 4</th><td>0.25784</td><td>0.42545</td><td>0.25080</td><td>0.39442</td></tr><tr><th>h = 5</th><td>0.32235</td><td>0.57081</td><td>0.31709</td><td>0.54217</td></tr><tr><th>h = 6</th><td>0.36468</td><td>0.72471</td><td>0.36028</td><td>0.66106</td></tr><tr><th>h = 7</th><td>0.41941</td><td>0.88183</td><td>0.39795</td><td>0.80009</td></tr><tr><th>h = 8</th><td>0.49479</td><td>1.03970</td><td>0.44024</td><td>0.93821</td></tr></tbody></table>

effective, however, it must be ambitious and quickly implemented: about $usd_{(2015)}$ 400 per t by 2050. Moreover, they find that such a world carbon tax must be accompanied by an equally ambitious public expenditure policy to enable the private sector to afford both the cost of mitigation and that of adaptation: public subsidies of at least 50% of the abatement costs are needed to avoid the financial tipping point. Fortunately, this expense never exceeds 10% of the current world gdp. It should provide the necessary incentives to reach net-zero emissions worldwide before 2075—even with the most pessimistic damage function—and to stay broadly around a +2°C temperature elevation by 2100. Their estimated global cost of the low-carbon transition is approximately usd 90 trillion up to 2035, which agrees with the Global Commission on the Economy and Climate (2018).

More than the details of these conclusions, which will have to be challenged as every scientific statement is, it is the coupling methodology between an emic and a nonlinear macrofinancial dynamics that, in our view, opens a new and promising avenue of research.

## Appeal of Lotka-Volterra models as an alternative foundation

Unsurprisingly, the predator-prey models generate highly correlated cycles in $\omega$ and $\lambda$. Grasselli and Maheshwari (2017)) and Grasselli and Maheshwari (2018) have shown that a slight modification of the Goodwin model enables these dynamics to fit quite convincingly employment and wage data of a dozen major OECD countries.

Back-testing their relative performance against a multivariate (var) specification, as reported by McIsaac (2021), yields overwhelmingly better results for both the Goodwin and the Van der Ploeg specification, for time horizons ranging up to 8 years ahead (see Fig. 8).

To our knowledge, McIsaac (2021) provides the first example of an economic model overperforming a var. This is quite a significant result: when back-tested, most standard macroeconomic models perform so poorly that most economists refrain from even testing them, to the point that a var is often viewed as the best-in-class modeling approach for short-term predictions. Indeed, the var often performs quite well for a 6-month to 2-year horizon simply because it relies on the linear extension of past trends: the inertia of aggregate economic variables often turns such a simple extension into the best predictive viewpoint. It is, therefore, all the more interesting that the nonlinearities involved in (8) succeed in overperforming a var: even in the very short run, and despite its
internal inertia, the intrinsic nonlinearity of economic dynamics matters.

## Conclusion

The new generation of iams we have introduced in this paper and the references therein provides new ways to think about climate policy. These insights can be partially summarized by looking back at Harrod-Domar's seminal equation (Harrod 1939) from the 1930s:

$$v = \frac{s}{g + \delta}, \quad (14)$$

where s designates the fraction of income that is saved (both by households and firms).

Harrod's infamous claim that 'capitalism' is intrinsically unstable was motivated by the following elementary observation: If each of these three variables, v, s, g = α + n, is exogenously given in the long-run, they can only verify (14) by a stroke of chance. In other words, Harrod's equilibrium is structurally unstable. Solow's main idea was to overcome this 'knife-edge' property by allowing some substitutability between capital and labor. As we have seen, this enables us to make v endogenous. The existence and global stability of the unique equilibrium in Solow's model were seen as evidence that the left-hand side of (14) could adjust so that Harrod-Domar's equation would always be fulfilled in the long run.

The family of prey-predatory models analyzed in this paper suggests that the situation is more subtle. On the one hand, (14) holds out of equilibrium in any of these models. Its fulfillment does not mean that an equilibrium has been reached. On the other hand, convergence towards a globally stable steady state obtains even with complementary production factors simply by adding a price field. Adding corporate debt also turns the Solovian equilibrium into a locally stable attractor. Finally, introducing substitutability eases its asymptotic local stability but, notwithstanding Solow, does not suffice to guarantee its global stability as soon as debts enter the picture (as they should). Other equilibria emerge. In particular, a Fisherian debt-deflationary steady state may compete with the Solovian-balanced growth path to attract the state of the economy. And an otherwise stable Solovian equilibrium can be destabilized by global warming, endogenous growth or corporate default with collateral. Therefore, substitutability and complementarity are 'not' the key drivers of 'capitalism's' potential instability.

Public policy, however, regains a new relevance inasmuch as it succeeds in steering the economy to the basin of attraction of a politically or ethically desirable attractor, despite the various 'forces' (such as global warming) that may pull it elsewhere. Comparative statics—the bread and butter of economics—should no longer consist of analyzing the local consequences, for a given equilibrium, of changes in policy or our environment, but in understanding how such changes might divert the economy from some equilibrium in favor of 'another one', or might lead to a bifurcation by which a long-run steady state gives way to a limit cycle or a strange attractor.

Another main takeaway of this paper is as follows. Given the damages that will most probably be inflicted on capital and gdp by global warming, we are presumably on the verge of a 'climate Minsky moment': the worldwide private sector might be drawn in the coming decades into a spiral of debt that will ultimately turn out to exceed its carrying capacity. Climate damages are shrinking

the stability valley of the economy, in addition to diminishing its overall performance.

In the same way, DICE models went further through Regional Integrated Climate-Economy (RICE), introduced in Nordhaus and Yang (1996), an avenue of research consisting of regionalizing idee models. This requires to introduce, for each region, an open dynamical core, economic trades and exchange rate dynamics. It should enable us to focus on regional co-stabilization phenomena or, on the contrary, crisis propagation. Such an extension will also enable the exploitation of the full potential of the coupling with a spatial climate model such as iloveClim. One key upgrade is the disaggregation of capital into multiple productive sectors, each affected differently by climate change. Even if we suppose that 87% of GDP production would not be directly impacted by climate change as first considered in Nordhaus (1991), the impact on one sector can destabilize the whole economy if there is little substitutability among intermediary consumption factors. Energy and agriculture, for example, are highly exposed to climate change, and while their contribution to GDP is small, the degradation of their efficiency would have a tremendous impact on the economy of an entire region.

The economic core and its climate extension presented here are also a complement to a growing literature on ecological SFC economic models of decarbonization such as reviewed in (Jacques et al. 2023), which will eventually lead to a systemic approach that should be able to integrate biophysical constraints, biodiversity loss, and the endogenization of eroi.

## Acknowledgements

We thank Nick Stern and Joe Stiglitz for insightful discussions on earlier versions of this work, as well as Simon Lebastard for his excellent technical assistance.

## References

Aguilar-Conraria, L. (2008) 'A Note on the Stability Properties of Good-wins Predator—Prey Model', Review of Radical Political Economics, 40: 518–23.

Akerlof, G., and Stiglitz, J. (1969) 'Capital, Wages and Structural Unemployment', The Economic Journal, 79: 269–81.

Arthur, W. B. (2021) 'Foundations of Complexity Economics', Nature Reviews Physics, 3: 136–45.

Ball, L. M., Leigh, D., and Mishra, P. (2022) Understanding U.S. inflation during the COVID era, Working Paper 30613, National Bureau of Economic Research, Massachusetts Avenue Cambridge, MA.

Barrage, L., and Nordhaus, W. (2023) Policies, projections, and the social cost of carbon: Results from the DICE-2023 model Technical report, Yale University.

Beinhocker, E. (2006) The Origin of Wealth: Evolution, Complexity, and the Radical Remaking of Economics. Brighton, Massachusetts: Harvard Business School Press.

Bottazzi, J.-M., and Hens, T. (1996) 'Excess Demand Functions and Incomplete Markets', Journal of Economic Theory, 68: 49–63.

Bovari, E., Giraud, G., and Mc Isaac, F. (2018a) 'Coping with Collapse: a Stock-Flow Consistent Monetary Macrodynamics of Global Warming', Ecological Economics, 147: 383–98.

Bovari, E., Lecuyer, O., and Mc Isaac, F. (2018b) 'Debt and Damages: What Are the Chances of Staying under the 2 C Warming Threshold?' International Economics, 155: 92–108.

Bovari, E., Giraud, G., and McIsaac, F. (2019) 'Financial Impacts of Climate Change Mitigation Policies and their Macroeconomic
Implications: a Stock-Flow Consistent Approach', Climate Policy, 20: 179-98.

Bügelmayer, M., Roche, D. M., and Renssen, H. (2015) 'Representing Icebergs in the iLOVECLIM Model (Version 1.0)—a Sensitivity Study', Geoscientific Model Development, 8: 2139–51.

Burke, M. et al. (2015) ‘Incorporating Climate Uncertainty into Estimates of Climate Change Impacts’, *Review of Economics and Statistics*, **97**: 461–71.

Cass, D. (1965) ‘Optimum Growth in an Aggregative Model of Capital Accumulation’, *The Review of Economic Studies*, **32**: 233–40.

Cass, D., and Stiglitz, J. (1970) ‘The Structure of Investor Preferences and Asset Returns and Separability in Portfolio Allocation: a Contribution to the Pure Theory of Mutual Funds’, *Journal of Economic Theory*, **2**: 122–60.

Champsaur, P., and Cornet, B. (1990) ‘Walrasian exchange processes’, in Gabszewicz, J., Richard, J. F., and Wolsey, L., (eds.) *Economic Decision Making: Games, Econometrics, and Optimisation*. pp. 1–13, Amsterdam: Elsevier Sciences Publishers.

Chiappori, P. A., and Ekeland, I. (1999) ‘Aggregation and Market Demand: an Exterior Differential Calculus Viewpoint’, *Econometrica*, **67**: 1435–57.

Claussen, M. et al. (2002) ‘Earth System Models of Intermediate Complexity: Closing the Gap in the Spectrum of Climate System Models’, *Climate Dynamics*, **18**: 579–86.

Davis, M. (2005) ‘Late Victorian Holocaust: El Niño Famines and the Making of the Third World’, *Renewable Agriculture and Food Systems*, **20**: 190–2.

De Vos, A., Biggs, R., and Preiser, R. (2019) ‘Methods for Understanding Social-Ecological Systems: a Review of Place-Based Studies’, *Ecology and Society*, **24**: 16–50.

Debreu, G. (1959) *Theory of Value: An Axiomatic Analysis of Economic Equilibrium*, New Haven, CT: Yale University Press.

Debreu, G. (1974) ‘Excess Demand Functions’, *Journal of Mathematical Economics*, **1**: 15–21.

Dhillon, A., and Mertens, J.-F. (1999) ‘Relative Utilitarianism’, *Econometrica*, **67**: 471–98.

Dietz, S., and Stern, N. (2015) ‘Endogenous Growth, Convexity of Damage and Climate Risk: how Nordhaus’ Framework Supports Deep Cuts in Carbon Emissions’, *The Economic Journal*, **125**: 574–620.

Elsner, W. (2004) ‘The "New" Economy: Complexity, Coordination and a Hybrid Governance Approach’, *International Journal of Social Economics*, **31**: 1029–49.

Evans, G. W., Guesnerie, R., and McGough, B. (2019) ‘Eductive Stability in Real Business Cycle Models’, *The Economic Journal*, **129**: 821–52.

Fang, F., and Sanglier, M. (1997) ‘Complexity and self-organization in social and economic systems.’, in *International Conference on Complexity and Self-Organization in Social and Economic Systems-Proceedings of the International Conference on Complexity and Self-Organization in Social and Economic Systems* (1994, Beijing, China), Lecture Notes in Economics and Mathematical Systems. Berlin, Heidelberg: Springer.

Folke, C., Colding, J., and Berkes, F. (2003) ‘Synthesis: Building resilience and adaptive capacity in social-ecological systems’, in Berkes, F., Colding, J., and Folke, C., (eds.) *Navigating Social-Ecological Systems: Building Resilience for Complexity and Change*. pp. 352–87. Cambridge, UK: Cambridge University Press.

Fostel, A., and Geanakoplos, J. (2008) ‘Collateral Restrictions and Liquidity under-Supply: a Simple Model’, *Economic Theory*, **35**: 441–67.

Geanakoplos, J., and Zame, W. R. (2014) ‘Collateral Equilibrium: a Basic Framework’, *Economic Theory*, **56**: 443–92.

Giraud, G., and Tsomocos, D. (2010) ‘Nominal Uniqueness and Money Non-neutrality in the Limit-Price Exchange Process’, *Economic Theory*, **45**: 303–48.

Giraud, G. et al. (2021) *Fossil assets: The new subprimes? Technical report Institut Rousseau, Paris*. https://institut-rousseau.fr/wp-content/uploads/2021/06/Rapport-actifs-fossiles-VF-FRLIGHT-1.pdf

Giraud, G., and Weiye, Z. (2023) ‘Stock-flow consistent macrodynamics with endogenous growth, collateral and default’, in *Environmental Justice Program* WP forthcoming, Georgetown University, Washington DC.

Global Commission on the Economy and Climate (2018) *Unlocking the Inclusive Growth Story of the 21st Century: Accelerating Climate Climate Action in Urgent Times*, Technical report The New Climate Economy, World Resource Institute, Washington DC.

Godley, W., and Lavoie, M. (2007) ‘A Simple Model of Three Economies with Two Currencies: the Eurozone and the USA’, *Cambridge Journal of Economics*, **31**: 1–23.

Goodwin, R. M. (1967) ‘A growth cycle: Socialism, capitalism and economic growth’, in *Essays in Economic Dynamics*. pp. 165–70, New York, NY: The MacMillan Press.

Goodwin, R. M. (1991) ‘Economic evolution, chaotic dynamics and the Marx-Keynes-Schumpeter system’, in Hodgson, G. M., and Screpanti, E., (eds.) *Rethinking Economics, chapter 9*. pp. 138–52, Cheltenham, UK: Edward Elgar Publishing.

Goosse, H. et al. (2010) ‘Description of the Earth System Model of Intermediate Complexity iLoveclim’, *Geoscientific Model Development*, **3**: 603–33.

Gordon, R. J. (2011) ‘The History of the Phillips Curve: Consensus and Bifurcation’, *Economica*, **78**: 10–50.

Gordon, R. J. (2013) ‘The Phillips Curve Is Alive and Well: Inflation and the Nairu during the Slow Recovery’, NBER Working Paper Series, 19390.

Grandmont, J. M. (1977) ‘Temporary General Equilibrium Theory’, *Econometrica: Journal of the Econometric Society*, **45**: 535–72.

Grasselli, M. R., and Maheshwari, A. (2017) ‘A Comment on “Testing Goodwin: Growth Cycles in Ten OECD Countries”’, *Cambridge Journal of Economics*, **41**: 1761–6.

Grasselli, M. R., and Maheshwari, A. (2018) ‘Testing a Goodwin Model with General Capital Accumulation Rate’, *Metroeconomica*, **69**: 619–43.

Hall, C. et al. (2001) ‘The Need to Reintegrate the Natural Sciences with Economics’, *Bioscience*, **51**: 663–73.

Harrod, R. (1939) ‘Essay in Economics Dynamics’, *The Economic Journal*, **49**: 14–33.

Jacques, P. et al. (2023) ‘Assessing the Economic Consequences of an Energy Transition through a Biophysical Stock-Flow Consistent Model’, *Ecological Economics*, **209**: 107832.

Kaldor, N. (1957) ‘A Model of Economic Growth’, *The Economic Journal*, **67**: 591–624.

Kalecki, M. (1968) ‘Trend and Business Cycles Reconsidered’, *The Economic Journal*, **78**: 263–76.

Keen, S. (1995) ‘Finance and Economic Breakdown: Modeling Minsky’s “Financial Instability Hypothesis”’, *Journal of Post Keynesian Economics*, **17**: 607–35.

Keen, S. (2021) ‘The Appallingly Bad Neoclassical Economics of Climate Change’, *Globalizations*, **18**: 1149–77.

Kemp, L. et al. (2022) ‘Climate Endgame: Exploring Catastrophic Climate Change Scenarios’, *Proceedings of the National Academy of Sciences of the United States of America*, **119**: e2108146119.

Keynes, J. M. (1936) *The General Theory of Employment, Interest and Money*, London: Palgrave Macmillan.
Knutti, R. et al. (2015) ‘A Scientific Critique of the Two-Degree Climate Change Target’, *Nature Geoscience*, **9**: 13–8.

Koopmans, T. (1963) *On the concept of optimal economic growth*
Cowles Foundation Discussion Papers 163, Cowles Foundation for
Research in Economics, Yale University, New Haven.

Krausmann, F. et al. (2013) ‘Global Human Appropriation of Net Primary Production Doubled in the 20th Century’, *Proceedings of the National Academy of Sciences*, **110**: 10324–9.

Levin, I., and Hesshaimer, V. (2000) ‘Radiocarbon: a Unique Tracer of Global Carbon Cycle Dynamics’, *radiocarbon. an international journal of cosmogenic isotope research*, **42**: 69–80.

Mantel, R. R. (1974) ‘On the Characterization of Aggregate Excess Demand’, *Journal of Economic Theory*, **7**: 348–53.

Martin, H. et al. (2023) ‘Extreme climate risks and the financial sustainability of adaptation’, in *Environmental Justice Program* WP, forthcoming, Georgetown University, Washington DC.

Matsumoto, A., and Szidarovszky, F. (2020) ‘Delay Growth Model Augmented with Physical and Human Capital’, *Chaos, Solitons and Fractals*, **130**: 1–9.

McIsaac, F. (2021) ‘Testing Goodwin with a Stochastic Differential Approach: the United States (1948–2019)’, *Metroeconomica*, **72**: 696–730.

Mertens, J.-F., and Rubinchik, A. (2017) ‘Discounting and Welfare Evaluation of Policies’, *Journal of Public Economic Theory*, **19**: 903–20.

Minsky, H. P. (1982) ‘The financial-instability hypothesis: Capitalist processes and the behavior of the economy’, in Kindleberger, C., and Laffargue, J. P., (eds.) *Financial Crisis: Theory, History and Policy*. pp. 13–39, Cambridge UK: Cambridge University Press.

Mora, C. et al. (2017) ‘Global Risk of Deadly Heat’, *Nature Climate Change*, **7**: 501–6.

Murphy, R. (2018) ‘William Nordhaus Versus the United Nations on Climate Change Economics’, *Econlib Articles*. https://www.econlib.org/library/Columns/y2018/MurphyNordhaus.html

Nakicenovic, N., Swart, R., and Alcamo, J. (2000) ‘Emissions scenarios. Technical report’, in *Intergovernmental Panel on Climate Change, Special report on emissions scenarios (SRES): a special report of Working Group III of the Intergovernmental Panel on Climate Change*. Cambridge: Cambridge University Press.

Nalewaik, J. (2016) ‘Non-linear Phillips Curves with Inflation Regime-Switching’, *FEDS Finance and Economics Discussion Series*, **2016**: 1–56.

Newel, K. (2018) ‘Evolving climate policy and Canada’s energy future’, in *Evolving Energy Realities: Adapting to What’s next*. 36th *North American Conference*, Cleveland, OH: International Association for Energy Economics.

Norberg, J., and Cumming, G. (2008) *Complexity Theory for a Sustainable Future*, New York, NY: Columbia University Press.

Nordhaus, W. D. (1991) ‘To Slow or Not to Slow: the Economics of the Greenhouse Effect’, *The Economic Journal*, **101**: 920.

Nordhaus, W. (1992) ‘The “DICE” model: Background and structure of a dynamic integrated climate-economy model of the economics of global warming’, in *Cowles Foundation Discussion Papers* 1009, *Cowles Foundation for Research in Economics*, Yale University, New Haven.

Nordhaus, W. (2013) ‘Integrated economic and climate modeling’, in Dixon, P. B., and Jorgenson, D. W., (eds.) *Handbook of Computable General Equilibrium Modeling*, volume 1. pp. 1069–13. Amsterdam: Elsevier.

Nordhaus, W. (2014) ‘Estimates of the Social Cost of Carbon: Concepts and Results from the DICE-2013R Model and Alternative Approaches’, *Journal of the Association of Environmental and Resource Economists*, **1**: 273–312.

Nordhaus, W. (2017) ‘Revisiting the Social Cost of Carbon’, *Proceedings of the National Academy of Sciences*, **114**: 1518–23.

Nordhaus, W. (2018) ‘Evolution of Modeling of the Economics of Global Warming: Changes in the DICE Model, 1992–2017’, *Climatic Change*, **148**: 623–40.

Nordhaus, W., and Boyer, J. (2000) *Warming the World: Economic Models of Global Warming*. Boston Massachusetts: MIT Press.

Nordhaus, W. D., and Yang, Z. (1996) ‘A Regional Dynamic General-Equilibrium Model of Alternative Climate-Change Strategies’, *The American Economic Review*, **86**: 741–65.

Petoukhov, V. et al. (2000) ‘CLIMBER-2: a Climate System Model of Intermediate Complexity. Part I: Model Description and Performance for Present Climate’, *Climate Dynamics*, **16**: 1–17.

Phillips, A. W. (1958) ‘The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1861-1957’, *Economica*, **25**: 283–99.

Pindyck, R. S. (2013) ‘Climate Change Policy: What Do the Models Tell us?’ *Journal of Economic Literature*, **51**: 860–72.

Pottier, A. (2014). “*L’économie dans l’impasse climatique : développement matériel, théorie immaterielle et utopie auto-stabilisatrice*”. 2014EHES0052. PhD thesis, EHESS. url: http://www.theses.fr/2014_EHES0052.

Pottier, A., and Nguyen-Huu, A. (2017) ‘Debt and Investment in the Keen Model: a Reappraisal of Modelling Minsky’, *Review of Keynesian Economics*, **5**: 631–47.

Preiser, R. et al. (2018) ‘Social-Ecological Systems as Complex Adaptive Systems’, *Ecology and Society*, **23**: 46–61.

Quignet, A. et al. (2018) ‘Online Dynamical Downscaling of Temperature and Precipitation within the iLOVECLIM Model (Version 1.1)’, *Geoscientific Model Development*, **11**: 453–66.

Ramsey, F. (1928) ‘A Mathematical Theory of Saving’, *The Economic Journal*, **38**: 543–59.

Ratner, D., and Sim, J. (2022) ‘Who Killed the Phillips Curve? A Murder Mystery’, *FEDS Finance and Economics Discussion Series*, **2022**: 1–36.

Robinson, J. (1974) ‘History Versus Equilibrium’, *Indian Economic Journal*, **21**: 202.

Roche, D. M. (2013) ‘$18^\circ$O Water Isotope in the iLOVECLIM Model (Version 1.0) – Part 1: Implementation and Verification’, *Geoscientific Model Development*, **6**: 1481–91.

Roche, D. M. et al. (2014) ‘Adding a Dynamical Cryosphere to iLOVECLIM (Version 1.0): Coupling with the GRISLI Ice-Sheet Model’, *Geoscientific Model Development*, **7**: 1377–94.

Romer, P. M. (1983) *Dynamic Competitive Equilibria with Externalities, Increasing Returns, and Unbounded Growth*. PhD thesis, The University of Chicago.

Ryzhenkov, A. (2022) ‘Why the Ramsey-koopmans-cass model underlying noticeable integrated assessment models is misleading in economic and climate projections’, in *World-Systems Evolution and Global Futures*. pp. 49–81. New-York: Springer.

Sage, R. F. (2019) ‘Global Change Biology: a Primer’, *Global Change Biology*, **26**: 3–30.

Saltzman, B. (1978) ‘A Survey of Statistical-Dynamical Models of the Terrestrial Climate’, *Advances in Geophysics*, **20**: 183–304.

Samuelson, P. A. (1966) ‘A Summing up’, *The Quarterly Journal of Economics*, **80**: 568–83.

Samuelson, P. A., and Solow, R. M. (1960) ‘Analytical Aspects of Anti-Inflation Policy’, *The American Economic Review*, **50**: 177–94.

Schlüter, M. et al. (2019) ‘Capturing Emergent Phenomena in Social-Ecological Systems’, *Ecology and Society*, **24**: 1–26.

Schoon, M., and Van Der Leeuw, S. (2015) ‘The Shift toward Social-Ecological Systems Perspectives: Insights into the Human-Nature Relationship’, *Natures Sciences Sociétés*, **23**: 166–74.

Solow, R. M. (1988) ‘Growth Theory and after’, *American Economic Review*, **78**: 307–17.
Stern, N. (2008) ‘The Economics of Climate Change’, *American Economic Review*, **98**: 1–37.

Stern, N., Stiglitz, J., and Taylor, C. (2022) ‘The Economics of Immense Risk, Urgent Action and Radical Change: Towards New Approaches to the Economics of Climate Change’, *Journal of Economic Methodology*, **29**: 181–216.

Sterner, T., and Persson, U. M. (2008) ‘An Even Sterner Review: Introducing Relative Prices into the Discounting Debate’, *Review of Environmental Economics and Policy*, **2**: 61–76.

Stiglitz, J. et al. (2016) ‘Agent-Based-Stock Flow Consistent Macroeconomics: Towards a Benchmark Model’, *Journal of Economic Dynamics and Control*, **69**: 375–408.

Stiglitz, J. et al. (2017) *Report of the High-Level Commission on Carbon Prices*, Washington, DC: The World Bank.

Taylor, L. (2021) *Reconstructing macroeconomics*. In *Reconstructing Macroeconomics*, Cambridge, MA: Harvard University Press.

Van der Ploeg, F. (1985) ‘Classical Growth Cycles’, *Metroeconomica*, **37**: 221–30.

Vidal, O. et al. (2017) ‘Global Trends in Metal Consumption and Supply: the Raw Material-Energy Nexus’, *Elements: An International Magazine of Mineralogy, Geochemistry, and Petrology*, **13**: 319–24.

Vidal, O. et al. (2019) ‘Prey-Predator Long-Term Modeling of Copper Reserves, Production, Recycling, Price, and Cost of Production’, *Environmental Science & Technology*, **53**: 11323–36.

Vines, D., and Wills, S. (2020) ‘The Rebuilding Macroeconomic Theory Project Part II: Multiple Equilibria, Toy Models, and Policy Models in a New Macroeconomic Paradigm’, *Oxford Review of Economic Policy*, **36**: 427–97.

Weitzman, M. L. (2010) ‘What Is the "Damages Function" for Global Warming and What Difference Might it Make?' *Climate Change Economics*, **1**: 57–69.

Weitzman, M. L. (2011) ‘Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change’, *Review of Environmental Economics and Policy*, **5**: 275–92.

Wells, J. (2012) *Complexity and Sustainability*, Routledge edn, London.

Werner, R. A. (2016) ‘A Lost Century in Economics: Three Theories of Banking and the Conclusive Evidence’, *International Review of Financial Analysis*, **46**: 361–79.

Woillez, M.-N., Giraud, G., and Godin, A. (2020) ‘Economic Impacts of a Glacial Period: a Thought Experiment to Assess the Disconnect between Econometrics and Climate Sciences’, *Earth System Dynamics*, **11**: 1073–87.

Zscheischler, J. et al. (2018) ‘Future Climate Risk from Compound Events’, *Nature Climate Change*, **8**: 469–77.