domain: sde qfin

text:
Matthieu Bordenave
email: .matthieu.bordenave@etu.utc.fr

Global bioeconomic SFC model for the study of financial instabilities *

Keywords: Bioeconomic model, agroforestry, climate damages, Goodwin, Keen, Stock Flow Consistent Dynamics, out of equilibrium JEL classification codes: C53, D72, E12, F64, Q23, Q41

This paper presents a global macroeconomic model that combines the economic impact of climate change with different agroforestry policies. The aim is to integrate climate damage and a dimension of biodiversity loss into an economic growth model, while stressing the importance of tropical reforestation policies for climate change mitigation. Using a Stock-Flow Consistent approach based on Goodwin-Keen logic, the non-linear monetary dynamics of underemployment and income distribution are modelled while taking into account the role of private debt as a factor of financial instabilities. The calibration at a global scale enables me to simulate different planetary scenarios.

Introduction

The 2015 Paris Agreement have politically crystallised the imperatives of a low-carbon economic transition. According to Global Commission on the economy and climate (2014), 90 trillion dollars would be needed globally to finance the infrastructure to achieve the zero emission target. In parallel with climatic considerations, the anthropogenic deterioration of natural environments reaches a critical state. The public consciousness of the impact of human activities on ecosystems can be partly attributed to [START_REF] Carson | Silent Spring[END_REF] with the publication of her famous book "Silent Spring". More contemporarily, the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES) regularly produces assessments of knowledge on biodiversity and ecosystem services and their interlinkages at the global level. According to their report [START_REF] Díaz | Résumé à l'intention des décideurs du rapport sur l'évaluation mondiale de la biodiversité et des services écosystémiques de la Plateforme intergouvernementale scientifique et politique sur la biodiversité et les services écosystémiques[END_REF], 75% of the Earth's environment is "severely altered" by human activities. Nearly one million animal and plant species are now threatened with extinction. These considerations underline the need to take biodiversity into account in models, since our economic system is vitally embedded into ecosystems.

In particular, healthy forest ecosystems play a significant role in the perpetuation of endemic species. They also contribute to the production of ecosystem services and climate mitigation. Carbon (C) stocks in organic matter constitute one of the planet's main carbon reservoirs. About 20-30% of anthropogenic CO2 emissions are directly related to the destruction of forests for agricultural purposes [START_REF] Houghton | Releases of carbon to the atmosphere from degradation of forests in tropical Asia[END_REF]. Bearing this in mind, Article 3 of the Kyoto Protocol states that afforestation, reforestation, and other forestry activities must be taken into account by stakeholders, including the United States, in meeting CO2 emission reduction targets IPCC (2000).

The zero emissions target generally assumes ambitious assumptions about carbon storage technologies and reforestation policies. Integrated assessment models (IAM) with an incorporated climate module had flourished this past decades [START_REF] Alkemade | GLOBIO3: A Framework to Investigate Options for Reducing Global Terrestrial Biodiversity Loss[END_REF][START_REF] William D Nordhaus | Projections and Uncertainties About Climate Change in an Era of Minimal Climate Policies[END_REF]Strapasson et al., 2020a; Elbehri and Food and Agriculture Organization of the United Nations, 2015). However, it has been shown that cost-benefit analysis are highly controversial for climate modelling [START_REF] Espagne | Disentangling the Stern/Nordhaus Controversy: Beyond the Discounting Clash[END_REF]. As an additional downside, general equilibrium models prevent the study of financial instabilities by design. IAMs had thus been developed in a more post-keynesian framework [START_REF] Dafermos | A stock-flow-fund ecological macroeconomic model[END_REF]; [START_REF] Bovari | Coping With Collapse_ A Stock-Flow Consistent Monetary Macrodynamics of Global Warming[END_REF]. Nevertheless, none of the heterodox IAMs includes an agroforestry module.

In this paper, I propose implementing global forest dynamics into "Coping with Collapse", a climate-economy model developed by [START_REF] Bovari | Coping With Collapse_ A Stock-Flow Consistent Monetary Macrodynamics of Global Warming[END_REF]. This modification allows for a more in depth discussion on the following economic and environmental questions: how will the world be able to finance the necessary climate transformations ? Which financial instability dynamics could these transformations lead to ? What is the role of forests in climate change mitigation? This paper is organized as follows: Section(2) briefly discuss the state-of-the-art in bioeconomic modelling and sets up our modelling framework, Section(3) present the results of our main scenarios. Section(4) draws a result overview, discusses the causal chain of propagation of climate policies, and presents sensitivity analysis elements. My main conclusions and areas for future research are outlined in the final section.

1 Modelling framework

Bioeconomic modelling

Many ecological and bioeconomic models are spatially explicit. They use a spatial grid of the soil to characterise it according to several criteria (land use, humidity, soil erosion etc.). Satellite imagery is most commonly used for this type of study, but it is also often based on empirical data. Once the soil is characterised, the model makes temporal projections according to several scenarios built on extrapolations or variable endogenisations. One spatially explicit model that is widely used in the research community is the RUSLE soil erosion model [START_REF] Panagos | Cost of agricultural productivity loss due to soil erosion in the European Union: From direct cost evaluation approaches to the use of macroeconomic models[END_REF][START_REF] Phinzi | The assessment of water-borne erosion at catchment level using GIS-based RUSLE and remote sensing: A review[END_REF]. The method generally consists of coupling RUSLE to a computational general equilibrium (CGE) model. They are multi-sectoral, i.e. they must have modules for each sector of the economy. This makes it possible, for example, to assess the market changes associated with a disturbance in the agricultural sector. In [START_REF] Panagos | Cost of agricultural productivity loss due to soil erosion in the European Union: From direct cost evaluation approaches to the use of macroeconomic models[END_REF]'s article, the losses calculated through the general equilibrium model are much lower than those calculated through the cost benefit approach. According to the authors, this difference is due to the philosophy of the general equilibrium economic model. They argue that when the agricultural sector contracts, factors of production are free to relocate to other sectors, thereby mitigating the overall loss of GDP. Typically in CGE models, these adjustments tend to be low cost and almost frictionless. In fact, CGE models represent an idealised, fully competitive economy. They therefore conclude that the estimated GDP losses should be considered as the lower bound of economic losses. Thus, the CGE philosophy will not be used in our model.

Another family of models is the calculators. These tools have been developed to make temporal projections of several scenarios. The aim is to assess the impact of the scenarios in terms of carbon trajectory, biodiversity, energy mix, etc. They exist on a global scale [START_REF] Strapasson | Modelling carbon mitigation pathways by 2050: Insights from the Global Calculator[END_REF] , on a European scale (European Environment Agency, 2018;[START_REF] Yu | Implications of decarbonizing the EU economy on trade flows and carbon leakages[END_REF][START_REF] Baudry | Mitigating GHG Emissions through Agriculture and Sustainable Land Use[END_REF]Strapasson et al., 2020a) or on a city scale [START_REF] Digiesi | A Carbon Footprint Calculator for the Municipal Waste Collection System of Bari[END_REF][START_REF] Dahal | Cities' Greenhouse Gas Accounting Methods: A Study of Helsinki, Stockholm, and Copenhagen[END_REF]. The main problem with these models is their multisectoral nature. The model is built in modules, requiring a sectoral and not an aggregated macroeconomic mode.

The last family of bioeconomic models is the integrated assessment models (IAM). They are built by combining different separate modules which aim to model different aspect of climateenvironment-economy system. While some simpler IAMs remain fairly abstract, the more complicated models can provide an impressive range of projections at a very high resolution, down to the levels of individual preferences, like electric vehicles or food regime. These IAMs are typically "optimal growth" models, maximising welfare over the long term. The core economic module used are general equilibrium or partial equilibrium approach [START_REF] Alkemade | GLOBIO3: A Framework to Investigate Options for Reducing Global Terrestrial Biodiversity Loss[END_REF] or (Elbehri and Food and Agriculture Organization of the United Nations, 2015, p.295-297)). Such IAMs exclude the possibility of crisis, degrowth due to overindebtness, mass unemployment or money endogenousity [START_REF] Giraud | The macrodynamics of household debt, growth, and inequality[END_REF].

Recent research has contributed to building alternatives to such IAMs by incorporating Keynesian features [START_REF] Barker | A new economics approach to modelling policies to achieve global 2020 targets for climate stabilisation[END_REF] or more post-Keynesian insights [START_REF] Dafermos | A stock-flow-fund ecological macroeconomic model[END_REF]. [START_REF] Berg | A stock-flow consistent input-output model with applications to energy price shocks, interest rates, and heat emissions[END_REF] and [START_REF] Fontana | Towards post-Keynesian ecological macroeconomics[END_REF] have studied environmental aspects of such stock-flow consistent models. To our knowledge, none of its alternative models integrates global agroforestry dynamics, even though the latter represents an opportunity for climate mitigation [START_REF] Raj | Climate Change and Agroforestry Systems: Adaptation and Mitigation Strategies[END_REF].

Overview of the model

The model is part of the IAM family, which stands for Integrated Assessment Model, i.e. it tries to link the main characteristics of the society and the economy. The main advantage of this kind of model is that they furnish an integrated system perspective [START_REF] Valeria | Evaluating integrated assessment models of global climate change[END_REF]. It provides a framework in which coupling models is possible. In this paper, I present a model born from the coupling of two existing models. The first one "Coping with collapse" was built by [START_REF] Bovari | Coping With Collapse_ A Stock-Flow Consistent Monetary Macrodynamics of Global Warming[END_REF], a model that combines the economic impact of climate change with the pivotal role of private debt. Its climatic module is inspired by the searching work of [START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF]. It introduces a negative feedback loop of temperature on GDP and capital. Contrary to the core economic model of DICE model [START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF], [START_REF] Bovari | Coping With Collapse_ A Stock-Flow Consistent Monetary Macrodynamics of Global Warming[END_REF]'s modelling approach is based on prey-predatory macrodynamics first introduced by Goodwin (1967) and then deepen by [START_REF] Keen | Finance and Economic Breakdown: Modeling Minsky's "Financial Instability Hypothesis[END_REF]. This stock-flow consistent modelling framework (SFC) is fitted to study financial-real interaction with multiple long-run equilibria. It is also characterized by its private debt and underemployment, endogenously determined.

The agroforestry module comes from Eriksson's FOR-DICE model. This module enables us to integrate the forest resource as a carbon sink and a source of emission through deforestation into our integrated assessment model. It incorporates three types of forest (boreal, temperate, tropical) as endogenous stocks of biomass in billion m 3 . Tropical forest is the only biomass stock affected by deforestation1 . However, each type of forest are affected by harvest through bioenergy and rounwood demands. Unlike the Eriksson's approach, the level of tropical deforestation and bioenergy harvest are not determined by solving an optimisation problem. The tropical deforestation is exogenous through different agroforestry scenarios, and bioenergy harvest level is determined by energy requirements for world production. Bioenergy harvest is also influenced by a cost minimisation of industrial energy budget constraint under several exogenous carbon tax scenarios. Public sector is not explicitly integrated in this IAM but public policies are materialized by a tropical deforestation control rate and different carbon price paths which impact the emission reduction rate and the share of fossil fuels in the energy mix. The model sequence is presented as following: Subsection(1.3) details the core economic module and the climate-related damages, Subsection(1.4) explicit the climate module and its associated carbon cycle, the Subsection(1.5)presents the agroforestry module which also include energetic equations and the last Subsection (1.6) show the stock-flow consistency of our model.

Economic module

Production, damage and abatement

The global production Y 0 (Eq.1) comes from the combination of capital K with constant capital to output ratio ν [START_REF] Inklaar | Capital, labor and tfp in pwt8[END_REF] and labor L with constant Harrod-neutral productivity growth rate α.

Y 0 = min( K ν ; aL) (1)
L = K νa (2) ȧ a = α (3)
The rise in temperature caused by the increasing amount of anthropic CO 2 e released into the atmosphere impacts the proper functioning of the economy through climate damages. Multiple phenomena are involved in economy's disturbance through, among others, sea level rise, ocean acidification, degradation of arable lands, frequent droughts or frequent storms, all with strong non-linear effects.

To integrate the latter effects into the economic dynamics, Nordhaus (2016) first introduces a convex damage function designed to express damages in terms of current output fraction. Nevertheless, the Nordhaus' damage function was criticized by [START_REF] Weitzman | Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change[END_REF] and [START_REF] Stern | The Structure of Economic Modeling of the Potential Impacts of Climate Change: Grafting Gross Underestimation of Risk onto Already Narrow Science Models[END_REF] because of it underestimation of climate risk. They have therefore proposed more convex polynomial functions which will be used in our model (Eq.4):

D = 1 - 1 1 + π 1 T + π 2 T 2 + π 3 T ζ3
(4) [START_REF] Dietz | Endogenous Growth, Convexity of Damage and Climate Risk: How Nordhaus' Framework Supports Deep Cuts in Carbon Emissions[END_REF] however, rightly noted that production factors themselves might be damaged by climate change. It might be interpreted as a reduced stock of capital. The latter is simulated thanks to (Eq.5):

D K = f K D (5) D Y = 1 - 1 -D 1 -D K (6)
With f K the share of capital damaged by climate such that f K ∈ {0; 1/3} according to [START_REF] Bovari | Financial impacts of climate change mitigation policies and their macroeconomic implications: a stock-flow consistent approach[END_REF].

The other phenomena triggered by climate change in the model is abatement effort. To reduce the burden of carbon tax2 , a fraction of current output A is redirected. It represents the intermediate cost of reducing CO 2 e emissions through abatement technology, and it depends also on the emission reduction rate chosen by the productive sector, i.e the fraction of production processes that is 'de-polluted'.

A = σp BS θ n θ (7) n = min{( p c p BS ) 1 θ -1 ; 1)} (8)
Note that σ latter involved is the carbon intensity of the economy3 and θ a parameter controlling the convexity of the cost.

The endogenously determined level of emission reduction rate n (Eq.8) derives from the cost minimisation between the abatement cost AY and the carbon tax p c E ind . Then, I subtract the cost of the agroforestry policy M C to the real output which is thus defined as following:

Y = Y 0 (1 -D Y )(1 -A) -M C (9)
The backstop technology p BS4 :

ṗBS p BS = δ p BS < 0 (10)
In contrast, carbon tax can follow different scenarios. They are treated as exponential trajectories:

ṗc p c = δ pcar () (11)
Note that with an initial value of 2 dollars per ton of CO2e in 2016 , p c (2100) = {50; 300} with respect to δ pcar = {0.04; 0.06}

Profit, investment and inflation

Profit Π (Eq.12) represents nominal output pY minus the costs of production, i.e. wages wL, debt interest rD, carbon tax pT f = pP c E ind and capital depreciation pδ d K:

Π = pY -wL -rD -pT f -pδ d K (12)
The profit share π is thus defined by:

π = Π/pY (13)
It is assumed that real investment, I, is determined by the profit share, π, which reflects the risk appetite of firms. The marginal cost of avoiding deforestation M C is subtracted to investment in order to take into account the cost of the chosen agroforestry policy 5 . The investments from which capital depreciation is subtracted thus determine the dynamics of capital.

I = κ(π)Y (14)
with κ() a bounded, increasing and linear function of the profit share π 6

K = I -δ D K (15)
Changes in nominal corporate private debt D depend on the difference between current nominal profit, Π, and investment pI, plus nominal dividends paid to shareholders, Π d (π) 7 :

Ḋ = pI + Π d (π) -Π -pδ D K (16) Π d (π) = ∆(π)pY (17)
Note that the equation (Eq.16) implies that firms can borrow money to finance dividends. This is not surprising, as the famous Modigliani-Miller theorem [START_REF] Hellwig | Bankruptcy, Limited Liability, and the Modigliani-Miller Theorem[END_REF] states that equity and debt are equivalent ways of financing a company's expenses. Moreover, contemporary oil companies are known to issue debt to pay their shareholders (see 2016, "Oil Majors Continue To Take On Debt to Pay Dividends").

The next formula (Eq.18) encapsulates the dynamics of inflation. The price of consumption, p, converges to its long-run equilibrium value through a delayed exponential adjustment with a relaxation time 1 η p . The long-run equilibrium price is given by a mark-up m multiplied by the average unit cost of production c:

i = ṗ p = η p (mc -1) (18)
And the average unit cost of production is, in our model, the wage payments:

c = wL pY (19)
5 MC as well as the various agroforestry policies will be defined in the Agroforestry Module

6 Namely κ(π) = κ 0 + kπ(π) with κ(π) ∈ [κ min ; κmax] 7 with ∆(π) = ∆ 0 + ∆π(π) with ∆(π) ∈ [∆ min ; ∆max]

Labour market

The dynamics of the global labour force are derived from the United Nations (2019) median scenario for 15-64 year olds. N, the global labour force, is thus assumed to grow along a sigmoid curve:

β(N ) = Ṅ N = q(1 - N N max ) (20)
with P N the upper limit of the population and q the speed of convergence to N max .

The employment (Eq.21) is determined by the ratio of employees L to the global available workforce N:

λ = L N (21)
The wage dynamics are derived from Philips curves with the functional form of Grasselli and Nguyen-Huu (2018) 8 . Workers negotiate their wages according to the employment rate:

λ λ = ϕ(λ) (22)

Climate module

The climatic module is inspired by the searching work of [START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF]. The negative feedback loop of temperature on GDP and capital introduced in Section(1.3.1) is possible thanks to the temperature dynamics embodied in this section. The rise on temperature comes from the rising anthropogenic CO2e emissions.

Emissions

The carbon emission E (Eq.23) are composed by emissions from each forest stocks (sequestration) EF n , ES emissions from manufactured forest products 9 and E ind industrial emissions (from energy conversion):

E = E E + n EF n + ES (23)
Industrial emissions EE, defined Section(1.5.4) are derived from the energy conversion needed to produce the level of total output. The amount of energy needed to produce Y 0 is defined as following, with σ, the carbon emission intensity of the economy 10 and n, the industrial mitigation efforts accounted by the emission reduction rate (defined Eq.8):

Energy = Y 0 σξ(1 -n) (24) σ σ = g σ (25) ġσ g σ = δ gσ (26)
where the production, Y 0 , leads to carbon emissions through the exogenous carbon ratio, σ.

The carbon emission output is declining over time due to an increase in carbon efficiency. This 8 with ϕ(λ) a linear function of employment rate, ϕλ = ϕ 0 + ϕ λ (λ). Note that the behavioral functions, κ() and ϕ(), have been bounded to avoid inconsistent behaviors that might fall far outside the estimation range. See [START_REF] Nguyen-Huu | Debt and Investment in the Keen Model: a Reappraisal of Modeling Minsky[END_REF] 9 EFn and ES defined section.1.5 with as mentionned above n = (BOR, T EM, T RO) 10 exogenously defined by gσ, ξ the energy parameter and δg σ , see [START_REF] Bovari | Coping With Collapse_ A Stock-Flow Consistent Monetary Macrodynamics of Global Warming[END_REF] exogenous technology change also implies that the energy efficiency increases over time. The carbon emissions from production are further reduced by the industrial mitigation efforts accounted by the emission reduction rate, n, which represents non-carbon-based technologies used to produce energy. These technologies include, for example, solar power, geothermal energy and nuclear power. The carbon emissions from production are converted back to energy units by the energy-emissions parameter.

Carbon cycle

The carbon cycle is composed of 3 communicating layers: the atmosphere (AT), the upper ocean and biosphere (UP) and the deep ocean (LO). Their dynamics are translated by the following system of differential equations:

   ĊO AT 2 ĊO U P 2 ĊO LO 2    =   E 0 0   + Φ   CO AT 2 CO U P 2 CO LO 2   (27)
With Φ, the matrix modelling carbon diffusion through the three layers:

Φ =   -ϕ 12 ϕ 12 C AT U P 0 ϕ 12 -ϕ 12 C AT U P -ϕ 23 ϕ23C U P LO 0 ϕ 23 -ϕ23C U P LO   (28)
where

C j i = C jp ind C ip ind , (i, j) ∈ {AT, U P, LO} 2 (29)
CO2e therefore diffuses from one layer to another in such a way that the relative pre-industrial concentrations C ip ind , i ∈ {AT, U P, LO} in each layer are respected at equilibrium. The accumulation of greenhouse gases changes the atmospheric chemical properties and thus the energy balance of this layer, triggering an increase in radiative forcing11 . The latter consists of the industrial radiative forcing F ind due to anthropogenic emissions and a residual forcing F exo12 :

F = F ind + F exo (30)
F ind = F 2xCO2 log(2) log CO AT 2 C AT p ind (31)
F 2xCO2 in Eq.( 31) represents the increase in radiative forcing resulting from a doubling of the pre-industrial CO2e concentration.

Temperature

The change in radiative forcing directly impacts the global average surface temperature T and the average deep ocean temperature T 0 :

C Ṫ = F -ρT -γ * (T -T 0 ) (32) C 0 Ṫ0 = γ * (T -T 0 ) (33)
with ρ, the radiative feedback parameter; γ * , the heat exchange coefficient between the two layers; C, the heat capacity of the atmosphere, land surface, and upper ocean layer; and C 0 , the heat capacity of the deep ocean layer.

The global thermal behaviour results from a coupled two-layer energy balance model that stands for: (i) the atmosphere, land surface and upper ocean with a mean temperature, T , and (ii) the deeper ocean with a mean temperature, T 0 . In this framework, the latter layer includes the long-term thermal inertia effects of the climate system. The mean atmospheric temperature change induces the two-frequency deviation responses: a transient climate response (TCR) and an equilibrium climate sensitivity (ECS)13 [START_REF] Geoffroy | Transient Climate Response in a Two-Layer Energy-Balance Model. Part I: Analytical Solution and Parameter Calibration Using CMIP5 AOGCM Experiments[END_REF].

Agroforestry module

The purpose of this module is to integrate the forest resource as a carbon sink on the one hand, and as a source of GHG14 emission/sequestration on the other. It is based on the model FOR-DICE developed by [START_REF] Eriksson | THE ROLE OF THE FOREST IN AN INTEGRATED ASSESSMENT MODEL OF THE CLIMATE AND THE ECONOMY[END_REF] whose neoclassical core DICE [START_REF] Nordhaus | An Optimal Transition Path for Controlling Greenhouse Gases[END_REF], [START_REF] Nordhaus | Warming the world: economic models of global warming[END_REF], [START_REF] William | A question of balance: weighing the options on global warming policies[END_REF], Nordhaus ( 2017)) is replaced and adjusted for our continuous post-Keynesian framework. It incorporates three forest types in the form of biomass stock15 S = (Boreal, T emperate, T ropical). The boreal and temperate forest stocks are affected by agroforestry harvesting (for the production of manufactured goods and bioenergy) while the tropical forest stock is also controlled by its deforestation rate16 .

Figure 1: Agroforestry module flowchart Green -Forest dynamics ; Red -Exogenous policies

Growth of forest stocks

In the FOR-DICE model, forest stocks are subject to growth as defined by the logistic growth equation of [START_REF] Clark | Mathematical Bioeconomics: The Optimal Management of Renewable Resources[END_REF]. Even if harvest decreases the biomass stock, it also increases its growth.

Indeed, an ageing forest, reaching its maximum carrying capacity, sees its growth rate decreases. That is why forest stock growth reaches its upper limit when the stock is at its minimum capacity. Note that the maximum carrying capacity F M AX S decreases with deforestation D S and is defined as:

Ḟ M AX S = - F M AX S F S D S (34)
with S ∈ (BOR, T EM, T RO). Since only the tropical forest stock is subject to deforestation, the maximum capacities of the temperate and boreal forest stocks are constant. The dynamics of forest biomass F S depends on the intrinsic growth of forest ψ S , the maximum carrying capacity Eq.( 34), harvest H S and deforestation D S :

ḞS = ψ S F S 1 - F S F M AX S -H S -D S (35)

Deforestation

Baseline exogenous GHG emissions from land in the DICE model [START_REF] William D Nordhaus | Projections and Uncertainties About Climate Change in an Era of Minimal Climate Policies[END_REF] are used as an approximation of the emissions due to tropical deforestation:

ĖT RO E T RO = δ E T RO (36)
These baseline emissions are converted to effective biomass deforested:

D T RO = E T RO θ T RO (1 -RD t ) (37)
with θ T RO , the tropical carbon intensity parameter 17 and RD, the control variable of deforestation 18 . From the deforestation rate RD t , I deduct the reduction of direct carbon emissions from deforestation due to the chosen agroforestry policy:

RE = E T RO RD t (38)
In order to estimate the financial impact that industries benefiting from deforestation will suffer in the case of reduced deforestation, a marginal cost function was calibrated by Mathilda Eriksson. However, this function was calibrated for a modelling time step of 10 years. It was therefore necessary to recalibrate it to follow the estimates made by [START_REF] Kindermann | Global cost estimates of reducing carbon emissions through avoided deforestation[END_REF] for our continuous framework:

M C = ϕ 1 e ϕ2RE (39)
This cost of avoiding deforestation is then taken into account in the output dynamic (Eq.9).

Harvesting

In our model, forest harvest H S is dedicated on the one hand to the production of bioenergy HB S19 and on the other hand to the production of manufactured goods HS:

H S = HB S + HS S (40)
Production of wood product HS is assumed to grow linearly following the global workforce growth:

ḢS = S χ S HSη L L + 1 -HS (41)
with χ S the share of boreal, temperate and boreal forest and η a preference parameter20 .

Energy

The equation( 24) provides us the energy equivalent for producing Y 0 . The carbon based energy is composed exclusively of fossil fuel carbon F O and bioenergy (forest biomass harvest intended for energy production) BI21 . This energy is modelled by a Cobb-Douglas function with constant returns to scale22 :

Energy = ζF O β BI 1-β (42)
where ζ is a scale parameter, β and 1 -β are respectively the elasticities of substitution of FO and BI. Note that fossil energies are limited resources:

F O max ≥ T t F O t (43)
BI, bioenergy is also modelled by a Cobb-Douglas function with constant returns to scale, it is composed of the biomass harvested HB S for each forest type with S = (T RO, BOR, T EM ) associated with their respective elasticities of substitution κ, Φ and 1 -κ -Φ and with ω a scale parameter:

BI = ωHB κ T RO HB Φ BOR HB 1-κ-Φ T EM
(44)

GHG emissions

The total greenhouse gas emissions are composed of:

• emissions from energy conversion E E (fossil and biomass);

• emissions from forest harvest and decomposition of wood products E S ;

• emissions from the change of biomass stock contained in the different types of forest E F ;

E E = F O + S HB S Θ S (45)
with Θ S a biomass-to-emissions conversion parameter for S ∈ (BOR, T EM, T RO).

Biofuels and fossil fuels release carbon immediately when burned. In contrast, manufactured wood products can store CO2 for a considerable time. This time will depend on the use of the wooden object in question. To describe the release of carbon in wooden objects, the half-life of products HL is commonly used. It reflects the average time it will take for half of the carbon in the manufactured products to be released. The decay equation:

δ CH = ln(2) HL (46)
represents the fraction of carbon released in each period. The amount of CO2 stored in wood products CH is modelled to decrease with the decay equation and to increase with the manufacture of new forest products Inf low:

Ċ H = -δ CH CH + Inf low (47) Inf low = ν S HS S Θ S (48)
ν represents the share of carbon from harvested wood that will go into long-lasting manufactured products. Much of the agroforestry biomass is lost in industrial processes23 . Thus, I can obtain the amount of carbon from industrial roundwood harvest:

E S = CHδ CH + (1 -ν) S HS S Θ S (49)
The emissions/sequestrations of forest biomass stocks are then worth:

E F S = (-ḞS -H S )Θ S (50)
By summing the three types of emissions EF , ES, EE, I obtain the total emissions (Eq.23).

Stock-flow consistency

Our model's stock-flow consistency is shown in the next page (1.6). Here, the accounting concept "investment Equals saving" is always true. Indeed, our economy's monetary counterpart may now be identified: M stands for total deposits, which equals M h , household deposits, plus M f , productive sector deposits.

Households own both types of equities, E f and E b , because dividends from both financial and non-financial businesses, are transferred to them. As the banks' financial balance is always zero, their equity, E b , can be safely assumed to remain constant. Similarly, I assume that the producing sector's equity has a constant market value (e.g., because stock markets are closed in this model).

In addition, it follows from Eq.( 16) and the accounting identity:

pY = Π + W + rD + pT f + pM C + δ D pK = pC + pI (51) that W + Π d + rD + pT f + pM C = Ḋ + pC (52) Thus Ṁ h = Ḋ = L -Ṁ f ,
h M c -M Loans -L c L c Equities E -E f -Eb Sum (net worth) X h X f = 0 X b = 0 X Transactions Current Capital Consumption -pC pC Investment pI -pI Acc. memo [GDP] [pY ] Wages W -W Capital depr. -(δ k + D K )pK (δ k + DK)pK Proactive policies pT f + pM C -pT f -pM C Int. on loans -r c L c r c L c Bank dividends Π b -Π b Productive sector dividends Π d -Π d Int. on deposits r M M h r M M c -r M M Column sum (balance) S h Π r -pI + (δ k + D K )pK

Results

Let us have a look at the numerical analysis up to 2100. To study the impact of agroforestry and public policies, two baseline scenarios have been chosen. Both of them implied the Weitzman (2012)'s damage function and their difference is based on the presence or absence of damage on the capital, namely f k ∈ {0; 1/3}) in Equation( 5). From these two frameworks, I will then be able to establish the climate impacts of different carbon tax trajectories and agroforestry policies while stressing possible associated economic instabilities.

BASELINE -extrapolative deforestation

In order to correctly interpret the economic impact of climate change into our model, it is necessary to show its steady growth equilibrium without damages and without agroforestry policy. An exogenous carbon price trajectory is implemented at an initial value of 2010 US$ 2 t/C0 2 -e and reaching 10 t/C0 2 -e in 2100. Then, climate damages are introduced on production (low damage) and on capital (high damage). The trajectory of the main economic variables is as follows: In the No damage case without carbon tax, the model converges to a Solovian steady state of growth. GDP increases exponentially as productivity also increases exponentially, reaching about USD 500 trillion in 2100, almost 10 times the current world output. The employment rate stabilises at around 70%, a result close to the OECD average. Inflation stabilises at around 5%. The ratio of private debt to GDP reaches around 150% in 2100, an average close to the current world powers24 . In short, without climate damage, the economic model reaches a stable growth equilibrium. Note that in this case, the amount of 130 GtC of GHG emissions is reached in 2100, i.e. more than three times the amount of annual emissions in 2015. These emissions imply an increase of 4.35 °C compared to the pre-industrial era.

However, in the absence of proactive public policies, it turns out that the model converges in the long run toward a "bad" attractor in both high and low climate damage cases. This climate-induced degrowth is characteristic of Fisher's debt deflation theory [START_REF] Fisher | The Debt-Deflation Theory of Great Depressions[END_REF]. The damages on production and the associated deflation inevitably impact the private debt to output ratio. Without policy implementation, the climate change is to create financial turmoil by the second half of the century.

In the baseline framework, the deforestation rate follows the exogenous path defined Equations (36;37). It is worth mentioning again that deforestation only takes place in tropical forests in our model. For the sake of clarity, the trajectory of the main variables of the agroforestry module is represented in the low damage case: In the baseline low damage case, with an extrapolative deforestation rate, the tropical stock of biomass is almost halved in 2100, namely 247.4 billion m 3 , supposing serious biodiversity losses. The sequestration capacity of the whole stock of forest (tropical, temperate and boreal ) reaches 3.33 GtCO2 thus giving a total emission of 112 GtCO 2 e at the end of the century. The temperature anomaly with the low carbon tax and given the latter amount of anthropogenic emissions is 4.23 C. Total harvest is growing due to the increasing bioenergy demand.

FOREST POLICY A -smooth reforestation

Now I implement a first agroforestry policy, namely a linearly increasing deforestation control rate with RD(2100) = 2 still in the low damage case. Here, deforestation is stopped around 2058 and then starts a process of reforestation until 2100. In addition to this policy, I want to study the impact of a more convex carbon tax (dotted lines)25 on the variables. The trajectory of the main variables is as follows: In presence of the smooth carbon policy, namely tax(2100) = 50$/tC, and the first forestry policy, the stock of tropical forest is equal to 110.81% of the 2015 stock, supposing a slight biodiversity recovery. The whole forest sequestration capacity reaches an amount of 5.48 GtC per year, which represent about twice the European Union emissions in 202026 . With these policies, the total amount of emissions is about 74 GtC0 2 e per year at the end of the century, approximately twice the amount of 2015. In comparison with the baseline scenario, the mean temperature anomaly is lowered by 0.2 °C, namely +4.02 °C in comparison with pre-industrial era mean temperature. Total harvest is growing due to the increasing energetic demand. There is a soft substitution from fossil energies towards bioenergy due to the carbon tax added to the fossil energies' price (see Appendix(A)).

By implementing a more convex carbon tax, namely tax(2100) = 300$/tC, financial stability marginally changes. The financial burden of the policy increases the private debt ratio of about 10% in comparison with the latter carbon tax. In both carbon policy cases, inflation is about 3% at the end of the century. Interestingly, the more convex carbon policy has adverse effects on forest stocks. There is a stronger carbon based energy substitution due to a higher energetic relative prices. The higher bioenergy demand thus implies a higher level of forest harvesting before 2096. In consequence, the 2100 stock of tropical forest is reduced to about 11% in comparison with the soft carbon policy case. However, in this case, there is a higher emission reduction rate due to the more convex carbon policy. This implies a strong substitution from carbon based energies toward non-carbon based energies (see Eq(24). In consequence, the demand for bioenergy27 , the demand for fossil energies and the level of carbon emission significantly decrease at the end of the century.

FOREST POLICY B -ambitious reforestation

Now I implement a more ambitious agroforestry policy, namely a linearly increasing deforestation control rate with RD(2100) = 2 still in the low damage case. Here, deforestation is stopped around 2038 and then starts a process of reforestation until 2100. In the same vein as the latter scenario, two carbon tax policies (dotted and non-dotted lines) are implemented. The trajectory of the main variables is as follows: In presence of the smooth carbon policy, namely tax(2100) = 50$/tC, and as a result of the reforestation policy, the stock of tropical forest is equal to 170% of the 2015 stock showing a great policy efficiency. This policy thus enhances the sequestration capacity of the forest stocks, namely 6.50 GtC/year28 . The effect of the reforestation policy in terms of mean temperature, in comparison with the extrapolative deforestation scenario, ceteris paribus, is about -0.3 °C. Here, the importance of such policy to alleviate climate change appears. Once again, the adverse effects of a more convex carbon policy (dotted lines) appear in this scenario, the carbon-based energy substitution implies a higher level of harvesting which in turns results in a lower stock of forest biomass.

The higher cost of this more ambitious agroforestry policy increases the burden of private debt ratio. In both carbon policy cases, the private debt to output ratio is higher than 600% at the end of the century. It turns out that the model converge to deflation in the long term. This result highlights the possibility of financial turmoil by the second half of the 21 th century if the economic growth our society is based on is not questioned. However, in the modelling framework, the carbon tax has a significant effect on the level of emission due to the associated emission reduction rate chosen by industries. Combining environmental and financial effects in a macroeconomic SFC model makes it possible to determine the sources and circumstances for future economic growth under different public policies. In the model, the carbon price trajectory is a tool that activates two levers: the emission reduction rate chosen by the industrialists on the one hand and the favouring of bio-energy over fossil fuels through the industrial minimisation of energy costs on the other. In addition, it is possible to study the consequences of different agroforestry policies on financial instabilities and carbon sequestration. With regard to all the scenarios, it appears that the + 1.5°C objective defined in the Paris Agreements seems to be strongly compromised. Indeed, even assuming a drastic reduction in emissions after 2050 until a quasi carbon-neutral society is reached in 2100, the temperature will probably reach more than 3.5°C after the pre-industrial era. According to the World Meteorological Organisation, it is likely that we will reach +1.5°C during the next decade 29 . Beyond the coupling of the economic and climate model with the agroforestry model, two scenarios emerged that had not been studied in the literature of this modelling framework: reforestation policies. These scenarios (see FOREST A and B, Subsection??) make it possible to understand the importance of implementing a global agroforestry policy. In the more ambitious scenario, reforestation leads to a carbon sequestration equivalent to 18% of current annual anthropogenic emissions by 2100. Nonetheless, a sole agroforestry policy would not be sufficient to attain the Paris Agreement goal. Interestingly, the model also shows possible adverse effects of a carbon tax on forest stocks. Indeed, by rising the carbon tax the change in energy relative price implies a shift from fossil fuel toward bioenergy which in consequence reduces the forest stocks.

Discussion

The climate-economy module used in this model has been studied in greater depth in [START_REF] Bovari | Financial impacts of climate change mitigation policies and their macroeconomic implications: a stock-flow consistent approach[END_REF]. They present the [0.25, 0.75] probability interval of the Monte Carlo simulations for the different damage functions. They study the probability distribution of the debt-to-output ratio and the temperature anomaly for the runs of all the parameter combinations in the different scenarios. They found that increasing the share of capital damage pushes up the private indebtedness. However, the effect on temperature appears to be weaker. Indeed, changes in the capital stock, through the growth engine, have only a small impact on emissions. They also found that "increasing the stringency of the policy mix by adding a subsidy for abatement technology allows the probability of achieving the Paris Agreement's target to be raised to 7%" (Bovari et al., 2020, p.193). [START_REF] Eriksson | Pricing forest carbon: Implications of asymmetry in climate policy[END_REF] provides a sensitivity analysis for the agroforestry module used in this model. In particular, she studied how the results change with different values of the intrinsic growth rate, the carrying capacity, the cost of avoiding deforestation and the energy elasticities.

Conclusion

This model allows for a better understanding of economic (growth, underemployment) and financial (private debt) interactions with climate instabilities. In this framework, it is clear that a global economic recession remains likely if we do not manage to decouple production from carbon emissions. Strong financial instabilities may arise through the gradual increase of private debt ratio and the spectrum of long run deflation. Thanks to the agroforestry module, it has been possible to show that a global reforestation policy remains a non-negligible solution when it comes to climate change alleviation. Coupled with a carbon tax, it enables to reduce significantly anthropogenic emissions. Nevertheless, I have shown that a carbon tax might imply adverse effects on biodiversity as it impacts negatively forest stocks.

In spite of these considerations, the model paves the way for further research. Apart from forest stocks, no other biodiversity indicators are included. A bird population dynamic, for instance, could be an interesting way to study how changes in land use are affecting wildlife. In addition, it would be possible to implement a feedback loop from forest health towards the economic module. Finally, other public policies may be worth implementing, such as a redistribution tool (e.g. a green subsidy) in parallel with the carbon tax. One could also imagine endogenising population dynamics by integrating family policies, for example.

A Appendix II -Calibration details

A.1 Energy demands and cost minimisation

In the model, and following [START_REF] Eriksson | Pricing forest carbon: Implications of asymmetry in climate policy[END_REF]'s methodology, the carbon energy needed to produce Y is composed of fossil fuels and bioenergy through the Cobb Douglas function:

Energy = ζF O β BI 1-β (53)
Constant returns to scale are assumed, i.e. energy varies in the same proportion as the factors of production used. This translates mathematically into the fact that the sum of the exponents is 1. I calculate β as the share of fossil energy in the energy mix in 2015. The total energy needed to produce Y in 2015 is 517 TJ. In the same year, 466.45 TJ of fossil energy were consumed. This gives me: The Cobb-Douglas energy function is now parameterised. It appeared interesting to impact the fossil energy demand with the carbon price. To do so, it is assumed that industrialists choose their fossil energy and bioenergy demand according to their budget constraint C 32 :

C = p F O F O + p BI BI (56)
By replace BI by its value corresponding to the Cobb-Douglas function, I obtain: As an example, in the no damage + deforestation case, the impact of the carbon tax on carbon based energy demands is as follows: The calculated value corresponds to the order of magnitude of market prices before the pandemic.

BI = E ζ 1-β F O -1 1 -β (

Figure 2 :

 2 Figure 2: Main variables for the BASELINE framework blueno damage ; goldlow damage ; redhigh damage

Figure 3 :

 3 Figure 3: Agroforestry module variables -BASELINElow damage yellow -Tropical ; green -Temperate ; light blue -Boreal

Figure 4 :

 4 Figure 4: Main variables -FOREST POLICY Alow damage yellow -Tropical ; green -Temperate ; light blue -Boreal

Figure 5 :

 5 Figure 5: Main variables -FOREST POLICY Blow damage yellow -Tropical ; green -Temperate ; light blue -Boreal

Figure 6 :

 6 Figure 6: Causal chain of propagation of carbon tax

  the FOR-DICE model of[START_REF] Eriksson | THE ROLE OF THE FOREST IN AN INTEGRATED ASSESSMENT MODEL OF THE CLIMATE AND THE ECONOMY[END_REF] the fossil energy demand FO is expressed in GtC and the bioenergy demand in m 3 . Thus, I calculate the calibration parameter ζ (T J.m3 -1.GtC -1) with FO 30 and BI 31 data from 2015:ζ = Energy F O β BI 1-β = 20.7239777065467(55)

30

  add the carbon tax to p F O and impact on demand: See IEA (2020) p.53 31 FAO 2015, see http://www.fao.org/faostat/en/#data/FO 32 With p BI ($/billion m³) and p F O ($/GtC) the respective prices of bioenergy and fossil fuels

Figure 7 :

 7 Figure 7: Carbon based energy demandsno damage left without carbon taxright with p car (2100) = 50$/tonC

  namely the change in company's debts is equivalent to the change in household savings.

		Households Productive sector	Banks	Sum
	Balance sheet
	Capital stock	pK		pK
	Deposits	M

Table 1 :

 1 Balance sheet, transactions and flows of fund

	Sb

Table 2 :

 2 Main variables in 2100

	SCENARIO		BASELINE	FOREST A		FOREST B
	2100 carbon tax	10 $/tonC	50 $/tonC	300 $/tonC	50 $/tonC	300 $/tonC
	Temperature		+4.23	+4.02	+3.81	+3.80	+3.58
	anomaly [°C]
	Sequestration		-3.33	-5.48	-5.76	-6.50	-6.86
	[GtC/year]
	Emission		112.05	74.01	5.73	61.14	1.19
	[GtC0 2 e/year]
	Tropical	forest	63.8	110.81	99.96	170.17	165.22
	stock [2015 %]
	Cost of forest policy	-	5.22	5.22	17.83	17.83
	[U S$tril.]
	Private debt ratio	1.49	2.06	2.24	6.79	11.56

Due to the fact that ongoing deforestation mainly impacts tropical forest(Murray et al.)

which depends on total anthrogenic CO 2 e emissions

which slightly declines over time, figuring out environmental efficiency of the economy

This can be translated as a new technology that produces a close substitute for an exhaustible resource using relatively abundant production inputs and renders the reserves of the exhaustible resource obsolete when the average cost of producing the close substitute falls below the spot price of the exhaustible resource[START_REF] Levy | From Hotelling to Backstop Technology[END_REF] is accessible at a declining price as time goes on

"The radiative forcing is the net change in the energy balance of the Earth system due to some imposed perturbation. It [. . . ] quantifies the energy imbalance that occurs when the imposed change takes place. Though usually difficult to observe, calculated RF provides a simple quantitative basis for comparing some aspects of the potential climate response to different imposed agents, especially global mean temperature, and hence is widely used in the scientific community. Forcing is often presented as the value due to changes between two particular times, such as pre-industrial to present-day, while its time evolution provides a more complete picture"(Myhre et al., 2013, p. 664)

The residual forcing results from various residual factors such as non-CO2-e long-lived greenhouse gases and other factors such as albedo changes, or the cloud effect. For simplicity, it is taken here as exogenous, as IPCC(Myhre et al., 2013, p. 681) showed it to be negligible and in line with representative concentration pathways. Here, Nordhaus (2016)'s representation is used: a linear trajectory up to 2100

The TCR represents the deviation at the end resulting from a linear doubling of the atmospheric CO2-e concentration, while the ECS accounts for the new equilibrium of the system, reached decades later due to its thermal inertia. Note that the TCR induced by our model is in line with IPCC results[START_REF] Myhre | Anthropogenic and Natural Radiative Forcing[END_REF]

greenhouse gas

in m 3 and therefore not spatially explicit

Indeed, most of the world's deforestation takes place in South America, Africa and South and South-East Asia; that is, tropical forests.[START_REF] Murray | Including International Forest Carbon Incentives in Climate Policy: Understanding the Economics[END_REF]

Note that RDt ∈ (0; 2). When RDt = 0, there is no control on deforestation rate. For RDt = 1, there is no more deforestation. If RDt > 1 starts the reforestation and when RDt = 2, the reforestation happens at a rate similar to the baseline deforestation rate.

defined Section1.5.4

Here assumed to be constant but could vary with changes in demand for different types of wood

A reasonable estimate as in 2018, 81.3% of the energy mix comes from fossil fuels and 9.3% from biomass(IEA, 2020)

Note that the carbon tax will impact the fossil fuel and bioenergy demand. See detail Appendix(A)

Approximately 36% of the original tree volume is converted into long-life products[START_REF] Ingerson | Wood Products and Carbon Storage[END_REF]

https://www.banque-france.fr/en/statistics/loans/debt-and-securities/ debt-ratios-intitutional-sectors-international-comparisons

namely tax(2100) = 300 $/tonC

See https://www.statista.com/statistics/450017/co2-emissions-europe-eurasia/

and accordingly the level of harvesting dedicated to bioenergy

Which represents more than half of 2020 China emission. See https://www.statista.com/statistics/ 239093/co2-emissions-in-china/

https://public.wmo.int/fr/medias/communiqu%C3%A9s-de-presse/selon-de-nouvelles-pr%C3% A9visions-du-climat-il-est-davantage-probable-que?fbclid=IwAR1vkxZh7-sjzQqAZBY1JAyore9Ok7XQoWVZ2kXcQRhAfEkm2vWZMOs1nAo

Acknowledgments

I thank the participants of the Georgetown Environmental Justice Program (GEJP) for giving me the opportunity to present this study to them and for their useful comments. I would like to give a special thanks to Antoine Godin, Paul Valcke, Gaël Giraud, Marc Lavoie, Louison Cahen-Fourot, Michaël Lainé and two anonymous referees for their fruitful comments and suggestions. All remaining errors are mine.

Funding

This work benefited from the technical support of the Agence Française de Développement and the financial support of the Sorbonne Université -Institut de la Transition Environnementale.

Disclosure statement

I declare that I have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

B Appendix III -Notation


Pierre Aboulker

Guillaume Aubian

Pierre Charbit

Stéphan Thomassé

P 6 , triangle)-free digraphs have bounded dichromatic number

The dichromatic number of an oriented graph is the minimum size of a partition of its vertices into acyclic induced subdigraphs. We prove that oriented graphs with no induced directed path on six vertices and no triangle have bounded dichromatic number. This is one (small) step towards the general conjecture asserting that for every oriented tree T and every integer k, any oriented graph that does not contain an induced copy of T nor a clique of size k has dichromatic number at most some function of k and T .

Introduction

In this paper, we only consider graphs or directed graphs (digraphs in short) with no loops, no parallel edges or arcs nor anti-parallel arcs (in particular our digraphs contains no cycle of length 2).

Given an undirected graph G, we denote by ω(G) the size of a maximum clique of G and by χ(G) its chromatic number. A class of graphs C is χ-bounded if there exists a function f , such that every graph G in C satisfies χ(G) ≤ f (ω(G)).

Given a graph (resp. a digraph) H, we denote by F orb ind (H) the class of graphs (resp. digraphs) that do not contain H as an induced subgraph (resp. induced subdigraph). A celebrated and still wide open question in the area of graph colouring is the following conjecture of Gyárfás [START_REF] Gyárfás | On ramsey covering-number[END_REF] and Sumner [START_REF] Sumner | Subtrees of a graph and chromatic number[END_REF] (see [START_REF] Scott | A survey of χ-boundedness[END_REF] for a survey on χ-boundedness).

Conjecture 1.1 (Gyárfás-Sumner) For any forest F , F orb ind (F ) is χ-bounded.

In this paper, we study an analogue of this conjecture for digraphs. For a digraph D we denote by ω(D) the clique number of the underlying graph of D and by #» χ(D) its dichromatic number, that is the minimum integer k such that the set of vertices of D can be partitioned into k acyclic subdigraphs. A class of digraphs C is #» χ -bounded if there exists a function f such that every digraph D in C satisfies #» χ(D) ≤ f (ω(D)).

Conjecture 1.2 (Aboulker, Charbit, Naserasr [START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF])

For any oriented forest

#» F , F orb ind ( #» F ) is #» χ -bounded.
It is enough to prove it for oriented trees (the proof is the same as for the undirected case, and can be found in [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF], Proposition 1.6). An oriented star is an oriented tree with at most one non-leaf vertex. Chudnovsky, Scott and Seymour [START_REF] Chudnovsky | Induced subgraphs of graphs with large chromatic number. XI. Orientations[END_REF] proved it for oriented stars as well as for two of the four possible orientations of the path on 4 vertices: →←← and ←→→ (they actually prove that for any integer k and any oriented graph H where H is either an oriented star or →←←, or ←→→, any digraph in F orb ind ( H) with clique number at most k can be partitioned into a bounded number of stable sets, which is clearly stronger). Cook, Masarík, Pilipczuk, Reinald and Souza [START_REF] Cook | Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P 4[END_REF] proved it for the two other orientations of the paths on 4 vertices: →→→ and →←→. Nothing more is known.

Proving the conjecture for directed paths is already a very challenging case. In this paper, we go a step further in this direction by proving the following, where #» P 6 denotes the directed path on 6 vertices.

Theorem 1.3 For every D ∈ F orb ind ( #» P 6 ) with ω(D) ≤ 2, #» χ(D) ≤ 382.
Note that we did not try to optimise the bound.

, # » T T k is a hero in F orb ind ( #» F ).
Additionally, a result of [START_REF] Harutyunyan | Two results on the digraph chromatic number[END_REF] implies that if H is not an oriented forest, then no digraph is a hero in F orb ind (H) except for K 1 (the digraph on one vertex) and # » T T 2 . In a seminal paper, Berger et al. [START_REF] Berger | Tournaments and colouring[END_REF] give a simple inductive characterization of heroes in the class of tournaments (these contain, of course, # » T T k , but many more). Note that if a class of digraphs C contains all tournaments, then a hero in C is in particular a hero in tournaments, but a hero in tournaments does not need to be a hero in C. Every class considered in the following contains all tournaments.

Let K k be the digraph on k vertices with no arc and observe that the class of tournaments is the same as F orb ind (K 2 ). Harutyunyan et al. [START_REF] Harutyunyan | Coloring dense digraphs[END_REF] extended the above result of Berger at al. by proving that, for every k ≥ 3, heroes in F orb ind (K k ) are the same as heroes in tournaments.

Following these works, a systematic study of heroes in classes of digraphs of the form F orb ind ( #» F ) where #» F is an oriented forest has been initiated in [START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF]. In particular, it is proved that if #» F is not a disjoint union of oriented stars, then the only possible heroes in F orb ind ( #» F ) are the transitive tournaments. On the other hand, it was conjectured in [START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF] that if #» S is a disjoint union of oriented stars, then heroes in F orb ind ( #» S ) are the same as heroes in tournaments, but this turned out to be false. In the paragraph below, we give a quick overview of the results on this particular question.

The result of Chudnovsky et al. [START_REF] Chudnovsky | Induced subgraphs of graphs with large chromatic number. XI. Orientations[END_REF] mentioned earlier implies that transitive tournaments are heroes in F orb ind ( #» S ) for any disjoint union of oriented stars #» S . In [START_REF] Aboulker | Heroes in oriented complete multipartite graphs[END_REF], it is proved that heroes in F orb ind ( #» P 3 ) are the same as heroes in tournaments. Denote by K 1 + # » T T 2 the disjoint union of K 1 and # » T T 2 and observe that

F orb ind (K 1 + # » T T 2 )
is the class of oriented complete multipartite graphs. Heroes in F orb ind (K 1 + # » T T 2 ) have been investigated in [START_REF] Aboulker | Heroes in oriented complete multipartite graphs[END_REF] where it is proved that they form a strict super class of transitive tournaments, and a strict subclass of heroes in tournaments (which disproved the aforementionned conjecture of [START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF]). Finally, heroes in F orb ind ( #» K 1,2 ) (where #» K 1,2 denotes the oriented star on three vertices with a vertex of out-degree 2, digraphs in this class are called locally-out tournaments) were studied in [START_REF] Aboulker | Decomposing and colouring some locally semicomplete digraphs[END_REF] and [START_REF] Steiner | On coloring digraphs with forbidden induced subgraphs[END_REF] (they are still conjectured to be the same as heroes in tournaments).

A digraph is t-chordal if all its induced directed cycle have length t. Surprisingly, for every t ≥ 3, the class of t-chordal digraphs has been proved [START_REF] Carbonero | Digraphs with all induced directed cycles of the same length are not #» χ -bounded[END_REF] to not be #» χ -bounded. Note that t-chordal digraphs are defined by forbidding an infinite number of digraphs, contrary to results mentioned above.

An oriented chordal graph is an orientation of a chordal graph. This is again a class of digraphs with a distinct flavour, obtained by taking all possible orientations of a class of (undirected) graphs. Heroes in oriented chordal graphs have been fully characterised in [START_REF] Aboulker | Heroes in orientations of chordal graphs[END_REF].

Definitions

If D be a digraph. We denote by V (D) its set of vertices and by A(D) its set of arcs. For

X ⊂ V (D) we define N + (X) = {y ∈ V (D) \ X, ∃x ∈ X such that xy ∈ A(D)} and N -(X) = {y ∈ V (D) \ X, ∃x ∈ X such that yx ∈ A(D)}. A subdigraph of D is a digraph obtained from D by
removing some arcs and some vertices (with all arcs incident to these vertices). If only vertices are removed, it is an induced subdigraph. For a given set of vertices X ⊆ V (D), we denote by D[X] the induced subdigraph obtained by removing V (D) \ X. Given a set of digraphs H, we say that a digraph D is H-free if it contains no induced subdigraph isomorphic to some member of H. We denote by F orb ind (H) the class of H-free digraphs. We say that D is triangle-free if ω(D) ≤ 2. Given a digraph H we say that D does not contain (or has no) H if D does not contain H as a (not necessarily induced) subdigraph.

We write x → y when xy ∈ A(D). A trail of a digraph D is a sequence of vertices x 1 x 2 . . . x p such that x i x i+1 ∈ A(D) for each i < p and each arc is used once (but vertices can be used several times). It is closed if x 1 = x p and its length is its number of arcs. We say odd closed trail for a closed trail of odd length. A trail (resp. closed trail) in which vertices are pairwise distinct is called a directed path (resp. directed cycle). The directed path of length k -1 is denoted by

#» P k . A k-dicolouring of D is a partition of V (D) into k sets V 1 , . . . , V k such that D[V i ] is acyclic for every i = 1, . . . , k. The dichromatic number of D, denoted by #» χ(D)
and introduced by Neumann-Lara [START_REF] Neumann-Lara | The dichromatic number of a digraph[END_REF] is the minimum integer k such that D admits a k-dicolouring. We will sometimes extend #» χ to subsets of vertices, using #» χ(X) to mean #» χ (D[X]) where X ⊆ V (D). For a set C of digraphs we write #» χ(C) to denote the maximum of #» χ (D) over all elements D in C, and write #» χ(C) = ∞ if this is not bounded.

3 Preliminaries

A set of vertices X is dipolar if for every x ∈ X, N + (x) ⊆ X or N -(x) ⊆ X.
This notion was first introduced in [START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF] under the name "nice set" and has been renamed "dipolar set" in [START_REF] Cook | Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P 4[END_REF]. The main tool using dipolar sets is the following lemma. We include its proof because it is short and enlightening for people unfamiliar with the dichromatic number. The key observation is that any directed cycle that intersects S and V (D) \ S intersects both S + and S -. Hence, by minimality of D, we can dicolour V (D) \ S with 2c colors. We can then extend this dicoloring to D by using colours 1, . . . , c for S + and c + 1, . . . , 2c for S -\ S + .

The strategy to prove our result is to show that every digraph in our class has a dipolar set with dichromatic number at most 191 and then apply Lemma 3.1. The next two results give simple techniques to bound the dichromatic number of a digraph, they will be extensively used to prove that the dichromatic number of some dipolar set is bounded. The first one is probably well known but we don't have any reference for it, the proof is very short. The next result is the dichromatic version of the celebrated Gallai-Roy-Vitaver theorem asserting that the chromatic number is upper-bounded by the largest size of a directed path. In a nutshell: the dichromatic number is upper-bounded by the largest size of a directed path of some feedback arc set.

Proposition 3.3

Let D be a digraph. Given a total ordering of the vertices of D, we say that an arc xy is forward if x precedes y in this ordering, and backward otherwise. The two following propositions are equivalent

• #» χ(D) ≤ k
• There exists an ordering of the vertices of D such that there exists no directed path on k + 1 vertices consisting only of backward arcs.

Proof : One direction is easy : if #» χ (D) ≤ k then there exists a partition (C1, C2, . . . C k ) of V (D) with Ci inducing an acyclic digraph. We construct an order on V (D) by putting all vertices of Ci before all vertices of Ci+1 for each i and within each class we use a topological sort. It is clear that in the resulting order, there can be no patch on more than k vertices where all arcs go backward since a backward arcs goes from one class to a previous one.

For the converse direction, assume that D has an ordering on its vertices such that there exists no directed path on k + 1 vertices consisting only of backward arcs and let us prove that D is k-dicolourable. For every x ∈ V (D), define f (x) the maximum number of vertices in a path consisting only of backward arcs and ending in x. By definition 1 ≤ f (x) ≤ k. Define Ci = f -1 (i) and let us prove that Ci does not contain any backward arc. Assume by contradiction xy is such an arc. Then there exists a path on i vertices ending in x consisting only of backward arcs, which implies that f (y) ≥ i + 1, contradiction. So each Ci induces an acyclic digraph, and thus #» χ (D) ≤ k.

The last lemma of this section is used to find induced directed paths. We will often use this lemma the following way :

• either a → b → c → d is an induced #» P 4 , • or b → c → d → a is an induced #» P 4 , • or a → b → c → d is a C 4 (in particular, a ∈ N + (C) ∩ N -(C)).
if a ∈ N + (C)\N -(C) (resp. a ∈ N -(C)\N + (C)),
then the first (resp. the second) output holds.

Proof of Theorem 1.3

For a subset X of vertices, we define recursively the sets N + k (X), N - k (X) and N k (X) by N + 0 (X) = N - 0 (X) = N 0 (X) = X, and for k ≥ 1 :

N + k (X) = N + (N + k-1 (X)) \ i<k N i (X) N - k (X) = N -(N - k-1 (X)) \ i<k N i (X) N k (X) = N + k (X) ∪ N - k (X)
We gather in the following claim several straightforward facts that we will use in the proof. Claim 4.1. For any X ⊂ V , the following hold

1. N + 1 (X) = N + (X), N - 1 (X) = N -(X) and N 1 (X) = N + (X) ∪ N -(X)
2. There are no arcs between X and N k (X) for k > 1.

If

x ∈ N k-1 (X), then either N + (x) ⊆ i≤k N i (X) or N -(x) ⊆ i≤k N i (X). 4. If x ∈ N + k (X) (resp N - k (X))
, there exists a directed path x 0 x 1 . . . x k (resp. x k x k-1 . . . x 0 ) such that x k = x and x i ∈ N + i (X) for every i ≥ 0. Items 1), 2) and 3) follow from the definition and 4) is easy to prove by induction on k.

Let now D be a triangle-free digraph in F orb ind ( #» P 6 ). Let C = x 1 x 2 . . . x 2k+1 x 1 be a (not necessarily induced) odd directed cycle of D of minimum length (we may assume it exists by Lemma 3.2). During the proof, for simplicity, we write

C for V (C), D[C] for D[V (C)] and N k (C) for N k (V (C)).
We are going to prove that the set

S = C ∪ N (C) ∪ N 2 (C) ∪ N 3 (C)
is dipolar and has dichromatic number at most 191, which implies Theorem 1.3 by Lemma 3.1.

Claim 4.2. S is dipolar. Moreover, #» χ (N 3 (C)) ≤ 2.
Proof. To prove that S is dipolar, we need to prove that for every vertex x in S, etiher

N + (x) or N -(x) is contained in S. Note that by Claim 4.1 item 3, this is trivial if x ∈ C ∪ N 1 (C) ∪ N 2 (C).
Assume now that x ∈ N + 3 (C) and let us prove that N + (x) ⊆ N (C) ∪ N 2 (C), which will imply both parts of the claim, since this proves that N + 3 (C) is an independent set. By Claim 4.1 item 4, there exists a directed path x 0 → x 1 → x 2 → x 3 , where x 3 = x and

x i ∈ N + i (C). If x 1 ∈ N + (C) \ N -(C)
, then, by Lemma 3.4, there exists a, b, c ∈ C such that abcx 1 is an induced #» P 4 . Since there is no arc between C and

N 2 (C) ∪ N 3 (C) (by Claim 4.1 item 2) and D is triangle-free, a → b → c → x 1 → x 2 → x 3 is an induced #» P 6 , a contradiction. So we can assume x 1 ∈ N + (C) ∩ N -(C). Consider y ∈ N + (x)
, and let us prove that y ∈ N (C) ∪ N 2 (C). Let t be an in-neighbour of x 0 in C and observe that t → x 0 → x 1 → x 2 → x 3 → y is a #» P 6 and the only way for it not to be induced (because of (Claim 4.1 item 2)) is that y is adjacent with one of {t, x 0 , x 1 }. If y is adjacent with t or x 0 , then y ∈ N (C). If y is adjacent with x 1 , and since

x 1 ∈ N + (C) ∩ N -(C), we get that y ∈ N 2 (C). We thus have proven that y ∈ N (C) ∪ N 2 (C). Similarly, if x ∈ N - 3 (C), then N -(x) ⊆ N (C) ∪ N 2 (C), which concludes the proof of this claim. Claim 4.3. #» χ(D[C]) ≤ 3.
Proof. By minimality of C, removing a vertex from C yields a digraph with no odd directed cycle, which thus has dichromatic number at most 2 by Lemma 3.2.

Claim 4.4. #» χ(N + (C) \ N -(C)) ≤ 4 and #» χ (N -(C) \ N + (C)) ≤ 4. Proof. Let us prove that #» χ (N + (C) \ N -(C)) ≤ 4.
We first prove that N + (x 1 ) ∪ N + (x 2 ) intersects all odd directed cycles of N + (C) \ N -(C). Suppose that it is not the case, and let C ′ be such an odd directed cycle. Let i ≥ 3 be minimum such that x i has an out-neighbour in C ′ (so that x 1 , . . . , x i-1 don't). Since C ′ ⊂ N + (C) \ N -(C), x i does not have an in-neighbour in C ′ , so by Lemma 3.4 applied to C ′ , there are 3 consecutive vertices a, b, c of C ′ , such that x i → a → b → c is an induced #» P 4 . By the choice of i, we then have that x i-2 → x i-1 → x i → a → b → c is an induced #» P 6 , a contradiction. Now, N + (C) \ N -(C) can be partitioned into two stable sets and a digraph with no odd directed cycle, and thus be 4-dicoloured. By directional duality,

#» χ(N -(C) \ N + (C)) ≤ 4. Claim 4.5. #» χ(N + 2 (C) \ N - 2 (C)) ≤ 2 and #» χ (N - 2 (C) \ N + 2 (C)) ≤ 2.
Proof. We prove that

#» χ(N + 2 (C) \ N - 2 (C)) ≤ 2.
Assume by contradiction this is not the case, so that by Lemma 3.2 we get an odd directed cycle Proof. The same proof works for the two assertions of the claim. Let ℓ ∈ {1, 2} and observe that, by hypotheses (triangle-free for ℓ = 1, or the assumption of the second sentence for ℓ = 2), for every x ∈ C, both N ℓ+ (x) and N ℓ-(x) are stable sets. Let X = (N ℓ+ (C) ∩ N ℓ-(C)) \ N ℓ ({x 1 , . . . , x 6 }). It is enough to prove that #» χ(X) ≤ 30 -12 = 18.

C ′ in N + 2 (C) \ N - 2 (C) . Let u be a vertex in N + (C) ∩ N -(C ′ ), which is non empty by definition of N + 2 (C). If u ∈ N + (C) \ N -(C),
For each vertex v ∈ X, choose (arbitrarily) a vertex x i (resp. x j ) in C such that there is a directed path of length l from v to x i (resp. from x j to v). Set out(v) = i and in(v) = j so that we define two functions out and in from X to {1, . . . , 2k + 1}.

In the case where ℓ = 2, let p

+ v (resp. p - v ) be a vertex such that v → p + v → x out(v) (resp. x in(v) → p - v → v). In the rest of the proof, v → p + v → x out(v) is understood as v → x out(v) in the case where ℓ = 1. For i ∈ [0, 5], let X i = {v ∈ X | out(v) = i mod 6} and then define X i,≥ = {v ∈ X i | out(v) ≥ in(v)} and X i,< = {v ∈ X i | out(v) < in(v)}. It is enough to prove that #» χ (X i,≥ ) ≤ 2 and #» χ(X i,< ) ≤ 1 for i = 0, . . . , 5.
So now i is fixed and we define a total order ≺ on X i the following way: we say first that u ≺ v when out(u) < out(v) and then extend arbitrarily this partial ordering to a total ordering of X i .

We first prove that #» χ(X i,≥ ) ≤ 2 using Proposition 3.3 applied to the reversal of ≺ defined above. Suppose then by contradiction that there exist a, b, c ∈ X i,≥ such that a ≺ b ≺ c and ab, bc ∈ A(D).

(c)), then x 1 → x 2 → • • • → x in(a) → p - a → a → b → c → p + c → x out(c) → • • • → x 2k+1 → x 1 is an odd closed trail (it does need to be a directed cycle because p - a = p + c is possible) and otherwise, x 1 → x 2 → • • • → x in(a) → p - a → a → b → p + b → x out(b) → • • • → x 2k+1 → x 1 is an odd directed cycle.
In both cases we get an odd directed trail that has strictly less vertices than C, and since an odd closed trail contains an odd directed cycle, we get our contradiction. Thus #» χ(X i,≥ ) ≤ 2.

We now prove that #» χ(X i,< ) ≤ 1. Suppose that there exist a, b ∈ X i,< such that b ≺ a and ab ∈ A(D).

Let

#» C 3,2 be the digraph with vertices u, v 1 , v 2 , w 1 , w 2 and arcs uv 1 , v 1 v 2 , v 2 w 2 , uw 1 , w 1 w 2 . Observe that if G ∈ F orb ind ( #» C 3,2 ), then for every x ∈ V (G), N + 2 (x) and N - 2 (x) are stable sets. Hence, by the previous claims (all of them), we get that for every triangle-free digraphs G ∈ F orb ind ({ We are now able to prove the last bit of the proof. Proof. By Claim 4.7, we may assume that N + 2 (C) ∩ N - 2 (C) contains #» C 3,2 as an induced subdigraph. Thus there exists u, v 1 , v 2 , w 1 , w 2 ∈ N + 2 (C) ∩ N - 2 (C) such that uv 1 , uw 1 , v 1 v 2 , w 1 w 2 , v 2 w 2 ∈ A(D). Moreover, there exists r, s ∈ C, and t ∈ N + (C) such that rs, st, tu ∈ A(D). Now, since r → s → t → u → v 1 → v 2 is not induced, t and v 2 are adjacent, and since r → s → t → u → w 1 → w 2 is not induced, t and w 2 are adjacent. Hence t, v 2 , w 2 forms a triangle, a contradiction.

#» P 6 , #» C 3,2 }), the set Q ∪ N (Q) ∪ N 2 (Q) ∪ N 3 (Q),
Altogether, we get that #» χ (S) ≤ 3 + 4 + 4 + 30 + 2 + 2 + 144 + 1 + 1 = 191, and thus #» χ(D) ≤ 382.

Lemma 3 . 1 (

 31 Lemma 17 in[START_REF] Aboulker | Extension of gyárfás-sumner conjecture to digraphs[END_REF]). Let C be a class of digraphs closed under taking induced subdigraph. Suppose that there exists a constant c such that each digraph D ∈ C has a dipolar set S such that #» χ(S) ≤ c. Then #» χ (C) ≤ 2c. Proof : Let D ∈ C be a minimal counter example, that is: #» χ (D) = 2c + 1 and for every proper subdigraph H of D, #» χ (H) ≤ 2c. By the hypothesis, D admits a dipolar set S, such that #» χ (S) ≤ c. Set S + = {x ∈ S | N -(x) ⊆ S} and S -= {x ∈ S | N + (x) ⊆ S}. By definition of a dipolar set, S = S + ∪ S -.

Lemma 3 . 2 .

 32 If a digraph D does not contain odd directed cycles as subdigraphs, then #» χ(D) ≤ 2.Proof : Let D be a digraph with no odd directed cycle and since the dichromatic number of a digraph is the maximum of the dichromatic number of its strong components, we can assume without loss of generality that D is strongly connected. In that case, we prove that the underlying graph G of D is in fact bipartite. Assume by contradiction G contains an odd cycle C = c1 → c2 → . . . → c 2k+1 → c1. For i = 1, . . . , 2k + 1, let Pi be a shortest directed path from ci to ci+1 (indices being taken modulo 2k + 1). Observe that either Pi = cici+1, or ci+1ci ∈ A(D), in which case Pi has odd length, for otherwise Pi ∪ {ci+1ci} is an odd directed cycle. Hence the union of the Pi for i = 1 . . . 2k + 1 forms a closed odd trail, which contains an odd directed cycle, a contradiction.

Lemma 3 . 4 .

 34 Let D be a triangle-free digraph, C a (not necessarily induced) odd directed cycle of D and a ∈ N (C). Then there exists consecutive vertices b → c → d of C such that

Proof:

  Assume a ∈ N -(C). Let us denote by x1, . . . , x 2k+1 the vertices of C (i.e. ∀i ≤ 2k, xixi+1 ∈ A(D) and x 2k+1 x1 ∈ A(D)). Assume without loss of generality that ax1 ∈ A(D). Let 1 ≤ p ≤ k be the maximum integer such that ax2p+1 ∈ A(D). Since the digraph is triangle-free, ax 2k+1 / ∈ A(D), so p ≤ k. It is straightforward to see that b = x2p+1, c = x2p+2, d = x2p+3 satisfies either the first or third item of the lemma. By reversing the arcs of the digraph, the same proof works if a ∈ N + (C).

  then by Lemma 3.4, there exist a, b, c∈ C such that a → b → c → u is an induced #» P 4 , which along with a vertex v ∈ N + (u) ∩ V (C ′ ) and the out-neighbour of v in V (C ′ ) forms an induced #» P 6, a contradiction (remember that by Claim 4.1 Item 2, there is no arc between C and C ′ ).Thus u ∈ N + (C) ∩ N -(C) and since V (C ′ ) is disjoint from N - 2 (C), u has no in-neighbour in V (C ′). Hence, by Lemma 3.4 applied on C ′ , there exist a, b, c ∈ V (C ′ ) such that u → a → b → c is an induced #» P 4 , which along with any v ∈ N -(u) ∩ C and the in-neighbour of v in C forms an induced #» P 6 , a contradiction.

Claim 4 . 6 .

 46 #» χ(N + (C) ∩ N -(C)) ≤ 30. Moreover, if for every x ∈ C, both N + 2 (x) and N - 2 (x) are stable sets, then #» χ (N + 2 (C) ∩ N - 2 (C)) ≤ 30.

  Since N ℓ-(x) is a stable set for every x ∈ C, out(a) = out(b) and out(b) = out(c) and thus out(c) ≥ 6 + out(b) ≥ 12 + out(a) ≥ 12 + in(a) If in(a) has the same parity as out(a) (and thus as out(b) and out

  Thus out(b) + 6 ≤ out(a) < in(a). If out(a) and in(a) do not have the same parity, then x out(a) → x out(a)+1 → • • • → x in(a) → p + a → a → p - a → x out(a) is an odd closed trail. Otherwise out(a) and thus out(b) have the same parity as in(a), and then x out(b)→ • • • → x in(a) → p - a → a → b → p + b → x out(b)is an odd directed cycle. In both cases it has strictly less vertices than C, a contradiction. Thus #» χ (X i,< ) ≤ 1 by Proposition 3.3.

  where Q is an odd directed cycle of G of minimum length, is dipolar and has dichromatic number at most 3 + 4 + 4 + 2 + 2 + 1 + 1 + 30 + 30 = 77. Hence, by Lemma 3.1 we get that: Claim 4.7. Triangle-free digraphs in F orb ind ({ #» P 6 , #» C 3,2 }) have dichromatic number at most 144.

Claim 4 . 8 .

 48 #» χ(N + 2 (C) ∩ N - 2 (C)) ≤ 144.

Acknowledgement

This research was partially supported by the ANR project DAGDigDec (JCJC) ANR-21-CE48-0012, by the ANR project Digraphs ANR-19-CE48-0013, and by the group Casino/ENS Chair on Algorithmics and Machine Learning.