Macroscopic approximation methods for the analysis of adaptive networked agent-based models: The example of a two-sector investment model
Jakob J. Kolb, Finn Müller-Hansen, Jürgen Kurths, Jobst Heitzig
MMacroscopic approximation methods for the analysis of adaptive networkedagent-based models: The example of a two-sector investment model
Jakob J. Kolb ∗ FutureLab on Game Theory and Networks of Interacting Agents,Potsdam Institute for Climate Impact Research, Potsdam, Germany andDepartment of Physics, Humboldt University Berlin, Berlin, Germany
Finn M¨uller-Hansen
Mercator Research Institute on Global Commons and Climate Change, Berlin, Germany andPotsdam Institute for Climate Impact Research, Potsdam, Germany
J¨urgen Kurths
Potsdam Institute for Climate Impact Research, Potsdam, Germany andDepartment of Physics, Humboldt University Berlin, Berlin, Germany
Jobst Heitzig
FutureLab on Game Theory and Networks of Interacting Agents,Potsdam Institute for Climate Impact Research, Potsdam, Germany (Dated: August 10, 2020)In this paper, we propose a statistical aggregation method for agent-based models with hetero-geneous agents that interact both locally on a complex adaptive network and globally on a market.The method combines three approaches from statistical physics: (a) moment closure, (b) pair ap-proximation of adaptive network processes, and (c) thermodynamic limit of the resulting stochasticprocess. As an example of use, we develop a stochastic agent-based model with heterogeneoushouseholds that invest in either a fossil-fuel or renewables-based sector while allocating labor on acompetitive market. Using the adaptive voter model, the model describes agents as social learnersthat interact on a dynamic network. We apply the approximation methods to derive a set of or-dinary differential equations that approximate the macro-dynamics of the model. A comparison ofthe reduced analytical model with numerical simulations shows that the approximation fits well fora wide range of parameters.The method makes it possible to use analytical tools to better understand the dynamical propertiesof models with heterogeneous agents on adaptive networks. We showcase this with a bifurcationanalysis that identifies parameter ranges with multi-stabilities. The method can thus help to explainemergent phenomena from network interactions and make them mathematically traceable.
I. INTRODUCTION
Agent-based modeling is a computational approach tosimulate systems composed of a large number of similarsub-units with many applications in ecology [1], business[2], sociology [3] and economics [4, 5]. ABMs are usedto study aggregate phenomena emerging from local in-teractions [6]. These interactions can be structured byspatial embedding of agents or by social networks [7–10].In economics, ABMs have been used to study for examplebusiness cycles [11], market power [4] and trade [5].ABMs are a promising alternative to dynamic stochas-tic general equilibrium (DSGE) modeling, the currentworkhorse of theoretical macroeconomics. DSGE mod-els usually build on the representative agent approach,i.e., they represent all individuals of one type such asfirms or consumers by one representative decision maker.The representative agent approach implies that theo-retical macroeconomics reduces macroeconomic phenom- ∗ [email protected] ena to assumptions about a few different representativeagents, leaving out many explanatory mechanisms forfluctuations in aggregate variables based on intra-groupinteraction and heterogeneity [12]. Furthermore, DSGEmodel often assume rational expectations, i.e., agentsknow the constraints and dynamics of the entire econ-omy, which has been criticised as philosophically unsoundand empirically unjustified [13]. But, due to these as-sumptions, most DSGEs allow for a thorough analyticalanalysis.ABMs allow implementing various individual decisionmodels that are behaviorally more realistic than full eco-nomic rationality. Agents are often assumed to be bound-edly rational and adapt their expectations, which is com-patible with the Lucas critique [14]. In ABMs, fluctua-tions in aggregate variables do not only arise from ex-ogenous shocks as in DSGE models but primarily fromirregularities in local interactions. Therefore, they offeran avenue for explaining various emergent phenomena[15] studied in empirical macroeconomics.On the other hand, ABMs are often very detailed sothat an analytic treatment is unfeasible. Therefore, inABMs, the difficulties arising from the aggregation of het- a r X i v : . [ ec on . T H ] A ug erogeneous and interacting agents are usually solved com-putationally. Because the model mechanisms are difficultto trace in the ‘black box’ of a computational model, theresults of ABMs are often difficult to interpret and cannotprovide mathematically sound proofs of relationships be-tween model variables. Results may therefore be difficultto generalize [16]. There has been some progress in thestandardization of model descriptions for ABMs [17], butthe lack of standardization, e.g. of decision rules, makesthe models difficult to compare [5, p. 239]. Even thoughthere are various techniques available for comprehensivemodel analysis [18], a systematic model exploration is un-common and mostly limited to sensitivity analysis withrespect to crucial parameters.Methods from theoretical physics have been appliedsuccessfully to various problems in economics for manyyears [19]. Here, aggregation methods from statisticalphysics can bridge the gap between analytic macroeco-nomic models such as DSGE approaches and agent-basedcomputational models [for a review of physics methods insocial modeling, see refs. 20, 21]. In contrast to macroe-conomic models, these approaches account for local inter-actions and use aggregation techniques to derive macro-dynamics, providing a true microfoundation of the result-ing macromodel. These kinds of approximation methodshave found much interest in the fields of financial eco-nomics, behavioral finance and evolutionary game theoryrecently and have produced interesting and promising re-sults, e.g. to explain macroeconomic fluctuations [e.g. 22]and understand propagation of financial shocks and theresulting systemic risk [e.g. 23].Many authors use mean field approximations to ag-gregate interactions between heterogeneous agents, e.g.making use of stochastic differential equations, Masteror Fokker-Planck equations [24–33]. Such approaches as-sume that each agent pair interacts with the same prob-ability. But many social and economic interactions arestructured and the structure can be described by com-plex networks [34]. To also capture the dynamics arisingfrom structured interactions, so-called moment closuremethods take the micro-structure of networks into ac-count when deriving macroscopic quantities [e.g. 35, 36].Thereby, they are able to show that often the networkstructure, whether fixed or evolving, has a crucial influ-ence on the dynamics not only quantitatively but alsoqualitatively in enriching the stability landscape and in-troducing additional (meta-) stable dynamical regimes,e.g. due to effects related to clustering and communitystructure.Yet, most of the literature regards either the networkbetween agents or the states of agents as static, implic-itly assuming different time scales for dynamics of andprocesses on the network. However, recent literature onopinion formation processes and the spreading of socialnorms in the field of computational social sciences sug-gests that both happen on a comparable timescale andcan therefore not be treated separately [7, 37]. For suchadaptive networks [7], moment closure techniques have been introduced in the physics literature to aggregate thefeedback between complex adaptive network dynamicsand dynamics of single node states [38–41]. Here, we in-troduce these techniques to economic modeling and com-bine them with approaches from macroeconomics whereinteractions also happen globally via aggregated vari-ables.The technical challenges of analytic approximationmethods for agent-based model has so far hampered theirwide-spread use in economics. But they have a hugepotential in providing profound insights into dynamicalproperties of economic systems: First, they help increas-ing performance of computer simulations, making cal-culation of single model runs much faster and thereforeallowing for a wider range of bifurcation and parameteranalyses. Second, in contrast to stochastic simulations,they make formal proofs of relations between macroscopicvariables possible. Third, they allow the derivation ofanalytical expressions of relations between model vari-ables from the dynamic equations, which is not possiblefrom single simulation runs. This paper makes a stepforward in showcasing how such methods can be used tocombine interactions on complex adaptive networks withmacroeconomic modeling. It is therefore a contributionto integrate non-standard behavioral assumptions intomacroeconomic models.The agent-based model we introduce as an illustrationof these methods is designed to investigate low-carbontransitions in an economy in the context climate eco-nomics and features both local interactions on a networkand system-level interaction through markets. We usean adaptive network approach for our model to demon-strate how the individual approximation techniques men-tioned above may be combined. In our model, the net-work of interactions between agents as well as the spread-ing of strategies between agents on this interaction net-work happen on a comparable timescale. In particular,we combine the different approximation techniques men-tioned above, namely moment closure, pair approxima-tion, and large system limit approximations to derive anaggregate description for the dynamics of our model [foran overview of the different techniques, see 42]. Themodel consists of heterogeneous households that inter-act and learn from neighbors on a social network anda two-sector productive economy. The households dif-fer in their investment strategy: they invest their savingseither in the “dirty” or the “clean” sector, each represent-ing a separate capital market through which the agentsinteract. Agents imitate the investment strategy of ac-quaintances that are better off with a higher probability.To the best of our knowledge this is the first study thatapplies such a combination of approximation methods ona model that combines structured local with global inter-actions of heterogeneous agents in a socioeconomic set-ting. By successfully applying approximation techniquesfor adaptive networks to our model, we demonstrate thatthey are useful for investigating economic relationshipswithin considerably complex models. Even though ourreference application is an economic one, this approxi-mation method can also be used to describe similarlystructured models in other fields of research such as socialecology, neuroscience or computational social science.In the remainder of the paper, we first describe thedetails of the model (Sec. II). Then, we derive an aggre-gate description of the model by applying three approxi-mation techniques, moment closure, pair approximation,and large system limit (Sec. III). We discuss commonal-ities and differences between computer simulations andthe approximation approach. Before concluding, we illus-trate how the derived macro-approximation can be usedin a bifurcation analysis to better understand the quali-tative properties of the non-linear model (Sec. IV). II. MODEL DESCRIPTION
To illustrate the use of the methods that we put for-ward, we develop a model of a stylized economy thatcaptures the shift from a fossil-fuel-based to a renewableenergy-based sector. Decarbonization pathways consis-tent with the Paris agreement require a rapid shift ofinvestments away from fossil fuel exploration and extrac-tion to the development and deployment of renewableenergies [43]. However, the implementation of climatepolicies is uncertain and expectations cannot be basedon self-consistent beliefs about the future. In conven-tional macroeconomic models such shifts can only occurdue to price signals either from improvements in greentechnology, increasing scarcity of fossil reserves, or car-bon pricing. While price signals are certainly important,movements advocating for the divestment from fossil fu-els point to the role of social norms and practices regard-ing investment decision to initiate and accelerate the en-ergy transition [44]. To better understand such culturallydriven situations of socioeconomic change, it is importantto develop models that can incorporate endogenous pref-erences [45, 46] and aspects of bounded rationality [47]such as imperfect foresight and information as well aslearning.Our model is designed to incorporate social dynam-ics that influence investment decisions [48, 49]. In thecontext of climate economics and policy, the literatureon social influence and norms has pointed out that suchmechanism are a leverage point to induce rapid changein socioeconomic systems [50–54]. The model focuses ontwo important mechanisms: First, investment strategiesare spread on a network, which can be understood asa social learning process [55] influenced by social norms[56]. Secondly, the network adapts endogenously basedon simple rules that model homophily [57, 58]. In the fol-lowing, we explain the different parts of our two-sectormodel in detail. Table I provides an overview of the vari-ables used for its formal description and Table II a list ofparameters.
Resource Stock
EconomicProduction (4,6) I n c o m e (11)(5) C a p i t a l , L a b o r (8,17) Clean (2,9)
Dirty (3,10) R e s o u r c e e x t r a c t i o n Households (15,16) L a b o r m a r k e t T e c h n o l o g i c a l L e a r n i n g (7,8)(1) FIG. 1. Schematic figure of the model consisting of two pro-duction sectors of which one depends on an exhaustible fossilresource stock as well as a set of heterogeneous householdsthat interact on an adaptive complex network and use sociallearning to decide upon which of two production sectors toinvest in. Boxes and bubbles denote modeled entities, arrowsdenote interactions. Numbers in brackets refer to equationsthat describe the specific part of the model.
A. Economic Production
Our model as outlined in Fig. 1 consists of two sectorsfor production and a set of heterogeneous households thatinteract via a complex adaptive social network. The twoproduction sectors employ different technologies. Theproduction technology in one sector depends on the inputof an exhaustible (fossil) energy resource R that is usedup in the process whereas the technology in the othersector does not. We call them the dirty and the clean sector accordingly. We assume that physical capital istechnology-specific and can not be reallocated betweenthe two sectors. Therefore, the heterogeneous householdsin the model provide different types of capital K j as wellas labor L to the sectors. We assume that the technologyin the dirty sector is fully developed and adequately de-scribed in terms of a fixed technological factor subsumedin the constant b d , the so-called total factor productivity.For fossil fuels, price elasticities of demand, i.e., changesin demand in response to increasing or decreasing prices,are low in real economies [59–61], even with the choicebetween alternative technologies factored in. We approx-imate this by assuming that the fossil resource cannotbe substituted by other production factors (capital, la-bor) in the dirty sector. This is in line with critique ofcommonly assumed substitutability of natural resourcesin some widely used production functions in neoclassicalmodels [62–66]. However, we acknowledge that a shift inthe output of economic production from manufacturingto services can lead to substitution of resources by capi-tal and labor [67] and argue that our model pictures thisin a shift of economic production from the dirty to theclean sector, which is described in the following.The clean sector represents a circular economy in whichthe output of final goods depends on the machinery,knowledge and effort used in its production and is notlimited by resource scarcity on the timescale under con-sideration. The technology C used in the clean sector isassumed to be still in development and is therefore ex-plicitly modeled. Following [68], we model technologicalprogress as learning by doing according to Wright’s law[69, 70]. We assume that C is proportional to cumulativeproduction but also depreciates with a constant rate χ .Depreciation can be regarded as a human capital effectthat leads to knowledge depreciation over time as in [71].This is also in line with the empirically observed decreasein learning rates for maturing technologies [68]˙ C = Y c − χC. (1)Capital, labor and technology/knowledge are assumedto be mutual substitutes. To satisfy these requirements,we use the following production functions: Y c = b c C γ L α c c K β c c , (2) Y d = min (cid:16) b d L α d d K β d d , eR (cid:17) , (3)Subscripts c and d denote the clean and dirty sector re-spectively, L c and L d are labor in the two sectors, α and β are elasticities of the respective input factors, b c and b d are the total factor productivities and K c and K d arethe capital stocks for the respective sector. Measuringunit production cost in the number of working hours asin the original study by [69], γ is equivalent the elastic-ity of learning by doing in the clean sector as outlined in[71].We assume an efficient usage of resources in the dirtysector, such that b d L α d d K β d d = eR (4)where 1 /e is the resource intensity of the sector, i.e., theamount of fossil resource needed for one unit of final prod-uct. The usage of the fossil resource R depletes a geologi-cal resource stock G with the initial stock G ( t = 0) = G :˙ G = − R. (5)In line with the assumptions common in the literature[72, 73], the cost of the fossil resource extraction andprovision c R depends on the resource flow R and theremaining fossil resource stock G such that ∂c R /∂R > ∂c R /∂G <
0. We chose the specific form to be c R = b R R ρ (cid:18) G G (cid:19) µ ; ρ ≥ , µ > , (6)such that at some point ∂Y d /∂R < ∂c R /∂R to take intoaccount that some part of the resource is not economic,i.e., its marginal cost exceeds its marginal productivity.We assume perfect labor mobility and competition for labor between the two sectors. This leads to an equilib-rium wage w that equals the marginal return for labor,i.e., the production increase from an additional unit oflabor: w = ∂Y c ∂L c = ∂Y d ∂L d − ∂c R ∂L d (7)with the sum of labor in both sectors equal to a constanttotal amount of labor: L c + L d = L. (8)As discussed before, we assume physical capital to bespecific to the technology employed such that it can onlybe used in the sector that it has been invested in origi-nally. This means that there are separate capital marketsfor the two sectors. We assume these capital markets tobe fully competitive resulting in capital rents equal tomarginal productivity, after accounting for energy costs: r c = ∂Y c ∂K c (9) r d = ∂Y d ∂K d − ∂c R ∂K d (10) B. Adaptive Network Model for InvestmentDecision Making
We model households as boundedly rational decisionmakers [74–76]: Households take their investment deci-sions, i.e., whether to invest their savings in the cleanor the dirty sector, not by forming rational expecta-tions [13, 14] but by engaging in social learning [55]to obtain successful strategies [77] with reasonable ef-fort. The outcomes of social learning crucially dependon the structural properties of the complex network ofsocial ties amongst the households [78]. The strong andstill increasing polarization of some societies on climatechange issues suggests that social dynamics reinforce op-posed positions in the population [79–84]. In static net-work models, such effects cannot be represented. There-fore, we model the adaptive formation of the social net-work endogenously. A well established principle for theemergence of structured ties in social networks is ho-mophily, i.e., the tendency that similar individuals getlinked [57, 85, 86]. The following model specification usessocial learning in combination with endogenous networkformation based on homophily to model the investmentdecisions of the households.We model N heterogeneous households denoted withthe index i as owners of one unit of labor L ( i ) = L/N and capital K ( i ) c and K ( i ) d in the clean and dirty economicsector respectively. Households generate an income I ( i ) from their labor and capital income which they use forconsumption F ( i ) and savings S ( i ) . The rate at whichhouseholds save their income is assumed to be fixed andis given by the savings rate s : I ( i ) = wL ( i ) + r c K ( i ) c + r d K ( i ) d , (11) F ( i ) = (1 − s ) I ( i ) , (12) S ( i ) = sI ( i ) . (13)A binary decision parameter o i ∈ [ c, d ] denotes the sectorin which the households decide to invest. As motivatedabove, we model decision making that is driven by twoprocesses: social learning via the imitation of successfulstrategies and homophily towards individuals exhibitingthe same behavior.We describe households as the nodes in a graph of ac-quaintance relations that change according to the follow-ing rules.1. Households get active at a constant rate 1 /τ .2. When a household i becomes active, it interactswith one of its acquaintances j chosen uniformly atrandom.3. If they follow the same strategy, i.e., they invest inthe same sector, nothing happens.4. If they follow a different strategy, i.e., they investin different sectors, one of two actions can happen:(a) Homophilic network adaptation: with proba-bility ϕ , the households end their relation andhousehold i connects to another household k ,that follows the same strategy.(b) Imitation: with probability 1 − ϕ , household i engages in social learning, i.e., it imitates thestrategy of household j with a probability p ji that increases with their difference in income.We follow previous results on human strategy updatingin repeated interactions from [77], when we assume theimitation probability as a monotonously increasing sig-moidal function of the relative difference in consumptionbetween both households: p ji = (cid:18) (cid:18) − a ( F ( i ) − F ( j ) ) F ( i ) + F ( j ) (cid:19)(cid:19) − . (14)As opposed to the absolute difference in the originalstudy by [77], the probability in our model depends onrelative differences. We set a = 8 to conform to their em-pirical evidence. This dependence on relative differencesin per household quantities is crucial for our method aswe will discuss later at the end of Sec. III D. We modelstrategy exploration as a fraction ε of events that arerandom, e.g., rewiring to a random other household orrandomly investing in one of the two sectors. Given thesavings decisions of the individual households, and as-suming equal capital depreciation rates κ in both sectors,the time development of their capital holdings is given by Symbol Variable description Y c , Y d clean and dirty production (flows) L c , L d labor employed in the clean and dirtysector K c , K d physical capital stocks of the cleanand dirty sector C clean technology R fossil resource use (flow) G fossil resource stock c R resource extraction cost w equilibrium wage r c , r d equilibrium capital rents in the cleanand dirty sector I ( i ) , F ( i ) , S ( i ) income, consumption expenses andsavings of individual i (flows) K ( i ) c , K ( i ) d individual capital stocks in the cleanand dirty sectorTABLE I. List of variables in the agent-based model. Allvariables are without units of measurement. ˙ K ( i ) c = δ o i c s (cid:16) r c K ( i ) c + r d K ( i ) d + wL i (cid:17) − κK ( i ) c , (15)˙ K ( i ) d = δ o i d s (cid:16) r c K ( i ) c + r d K ( i ) d + wL i (cid:17) − κK ( i ) d , (16)where δ ij is the Kronecker Delta. The total capital stocksin the two sectors are made up of the sum of the individ-ual capital stocks K j = N (cid:88) i K ( i ) j = N k j , (17)where k j is the average per household capital stock of agiven capital type.We acknowledge the fact that different model specifica-tions are possible and interesting. For instance, we onlyconsider fixed savings rates and the decision between twocapital assets and leave the analysis of the interestingpossible effects of households setting their savings ratesindividually to another study [87]. However, we want topoint out that the approximation methods that we de-velop in the following are highly useful to gain insightsfrom different but similar models that rely on complexadaptive interaction networks. C. Numerical Modelling and Results
With the model specifications from Sec. II, theparametrization in Tab. II and appropriate initial con-ditions for the dynamic variables, the model can be sim-ulated numerically. For this, we implemented the dynam-ics in the multi-purpose programming language Python.
Symbol Value Parameter description N
200 Number of households M b c
1. Total factor productivity in the cleansector b d
4. Total factor productivity in the dirtysector b R .1 Initial resource extraction cost e κ χ γ α c α d β c β d ϕ /τ
1. Rate of opinion formation events ε G L
100 Total labor s ρ µ The implementation of the ABM as well as the numeri-cal analysis using the approximation methods describedin the following are available on the github software ver-sioning service in [88]. In the following, we discuss theresulting aggregate dynamics.Figure 2 displays an exemplary average evolution ofour model calculated as the mean of 100 simulation runs.The simulation starts with initial conditions of abundantfossil resources G and low clean technology knowledgestock C (panel b) as well as equally low capital stocks inthe clean and dirty sector K c and K d (panel c). As weshow later (see Sec. IV), the rest of the initial configu-ration of the model is rather irrelevant for the selectedparameter values listed in Tab. II, since there is only onestable dynamical equilibrium as long as resource extrac-tion costs are negligibly low. The high initial capital rents r c and r d are a direct result of our model assumptions andinitial conditions, more precisely, the assumption thatcapital rent equals marginal productivity in Eq. 9 and 10 and that of decreasing marginal productivity due toour choice of β i in combination with the initial conditionof low capital and a fixed labor supply. Also as a di-rect consequence of these assumptions, the capital rents r c and r d decrease over time as the capital stock is builtup. Initially (from t = 0 to t = 100), as a result of ourchoice of total factor productivities b i and due to lowfossil resource extraction costs, capital productivity (andtherefore capital rent r ) is higher in the dirty sector thanthe clean sector (see panel a). Consequently, the major-ity of households invest in the dirty sector which leadsto a high capital stock K d (panel c) and high productionoutput Y d (panel d) in this sector.Regarding the capital rents, we would expect the sys-tem to move towards a dynamic equilibrium in which thecapital rent is equal in both sectors, i.e., r d = r c , if ev-erything else remained constant. However, we find thatthere is a persisting difference between r c and r d between t = 50 and t = 100. This difference can be explained bythe exploration of investment strategies even if they per-form worse, which brings the shares of clean and dirtyinvestors closer together. In terms of the depicted vari-ables this means that it brings n c closer to 0 . t >
100 the depletion of the fossil resource leads tosignificantly increasing resource extraction costs. Con-sequently, the marginal productivity of dirty capital K d decreases and so does r d , leading to a peak in accumula-tion of capital in the dirty sector around t = 100 (panelc). Once the relative return on capital in the clean sectorincreases, households start to adopt a clean investmentstrategy visible in an increase in n c in panel a. When thefossil resource stock reaches its economically exploitableshare at around t = 200, the overall productivity in thedirty sector reaches zero, leading to full employment ofall available labor in the clean sector. This drives de-mand for capital in the clean sector up, accelerating thechange from dirty to clean investment. As all householdsexcept for the share caused by exploration are investingin the clean sector, the system reaches an equilibriumwith high capital in the clean sector and low capital inthe dirty sector.Notably, we find an increasing variance in the frac-tion of households investing in the clean sector beforeand around the transition, which means that due to thestochasticity of the social learning process the transitionhappens earlier for some simulation runs than for others.Nevertheless, we find that the inertia of the model result-ing from the large accumulated stock of capital that isspecific to the dirty sector eventually leads to an almostentire depletion of the fossil resource.The adaptation dynamics in our model can lead to afragmentation of the network with stark economic con-sequences. As results in Appendix B show, an increasedrewiring rate ϕ in the network adaptation process leadsto a strongly delayed shift of investment from one sec-tor to the other during the transition, even though theincentive in terms of an increased return r c for the invest-ment in this sector is high. This fragmentation is equiv- r a) r c r d n c n c t k n o w l e dg e s t o c k C ×10 b) C G r e s o u r c e s t o c k G ×10 t e c o n o m i c p r o d u c t i o n Y ×10 d) Y c Y d c a p i t a l s t o c k K ×10 c) K c K d FIG. 2.
Example trajectory of the ABM.
Solid lines show mean results from 100 runs of the model. Grey areas aroundsolid lines show their standard deviation. The panels show capital rents in the clean and dirty sector r c and r d as well as thefraction of households investing in the clean sector n c in panel (a), knowledge and resource stock C and G in panel (b), outputof clean and dirty sector Y c and Y d in panel (c) and capital stocks K c and K d in the clean and dirty sector in (d). Initialconditions are G = G , C = 1, K ( i ) j = 1 for the economic subsystem. For the investment decision process, the initial opinions ofthe N = 200 households are drawn from a uniform distribution. Their initial acquaintance structure is an Erd˝os-Renyi randomgraph with mean degree k = 10. alent with a strong decline in the fraction of active edgesin the network, e.g. the fraction of edges that connecthouseholds investing in different sectors of the economy.This finding is consistent with a major result of adap-tive network modeling studies that show that adaptationwill lead to fragmentation of a network at high rewiringrates ϕ Do and Gross [38], Min and Miguel [41], Gross et al. [89], B¨ohme and Gross [90]. Such network proper-ties emerging from adaptation dynamics have been stud-ied for example in the context of opinion dynamics, epi-demics and social-ecological systems [7, 40, 91, 92]. Onecould suspect that the slow-down in the transition fromone sector to the other results from the decreased rateof imitation events as their frequency scales with 1 − ϕ .However, the results in Appendix A show that this effectis particular to the adaptive network model and cannotbe reproduced in a well-mixed system simply by adjust-ing for the reduced frequency of imitation events. Ap-pendices B and A discuss further differences between thefull model and special cases without adaptation as wellas well-mixed interaction. III. APPROXIMATE ANALYTICAL SOLUTION
Structurally, the model described in Section II con-sists of a set of coupled ordinary differential equations(1), (5), (15) and (16) with algebraic constraints (4), (7),(8), (9) and (10) for the economic production processand a stochastic adaptive network process for the sociallearning component that is described by the rules 1 to 4bin Section II B. The state space of this combined processconsists of two degrees of freedom of the knowledge stockand the geological resource stock as well as 2 N degreesof freedom for the capital holdings of the set of all in-dividual households plus the configuration space of theadaptive network process of the social learning compo-nent. We denote the variables of this process by capitalletters ( C, G, K ( i ) j . . . ). To find an analytic descriptionof the model in terms of a low dimensional system ofordinary differential equations, we approximate it via aPair Based Proxy (PBP) process, a stochastic process interms of aggregated quantities, thereby drastically reduc-ing the dimensionality of the state space. We denote thevariables of this process with capital letter with bars ( ¯ X ,¯ Y , ¯ Z , ¯ K ( k ) l . . . ).The derivation of this approximate process is done inthree steps: First, we solve the algebraic constraints tothe economic production process given by market clear-ing in the labor market and efficient production in thedirty sector – loosely following [93]. Second we use a pairapproximation to describe the complex adaptive networkprocess of social learning in terms of aggregated vari-ables, similar to [91]. Third, we use a moment-closuremethod to approximate higher moments of the distribu-tion of the capital holdings of the heterogeneous house-holds by quantities related to the first moments of theirdistribution.Finally, we take the limit of infinitely many households(large system- or thermodynamic limit) to obtain a de-terministic description of the system. A. Algebraic Constraints
To calculate labor L c and L d as well as wages in the twosectors, we use equations (6) and (7) and for simplicityassume ρ = 1 and µ = 2. We also assume equal laborelasticities in both sectors α d = α c = α resulting in w = ∂Y d ∂L d − ∂c R ∂L d = ∂Y d ∂L d − ∂c R ∂R ∂R∂L d = ∂Y d ∂L d − ∂c R ∂R ∂∂L d Y d e = ∂Y d ∂L d − b R G G ∂∂L d Y d e = b d αL α − d K β d d (cid:18) − b R e G G (cid:19) (18)for the dirty sector and w = b c αL α − c K β c c C γ (19)for the clean sector. Combining these results via equation(8) and substituting X c = ( b c K β c c C γ ) − α , X d = ( b d K β d d ) − α ,X R = (cid:18) − b R e G G (cid:19) − α (20)and solving for w yields: w = αL α − ( X c + X d X R ) − α . (21)Plugging (21) into equations (18) and (19) results in L c = L X c X c + X d X R , (22) L d = L X d X R X c + X d X R (23)for labor in the two sectors, and plugging this into (4)leads to R = b d e K β d d L α (cid:18) X d X R X c + X d X R (cid:19) α (24) for the use of the fossil resource. Using the results for L c and L d together with equations (9) and (10), the returnrates on capital result in r c = β c K c X c L α ( X c + X d X R ) − α , (25) r d = β d K d ( X d X R ) L α ( X c + X d X R ) − α . (26)It is also worth noting that if we assume constant re-turns to scale with respect to capital and labor, e.g., β c = β d = 1 − α, (27)(even though it is not necessary for our method) thisyields zero profits in both sectors: Y c = wL c + r c K c ,Y d = wL d + r d K d + c R . To sum up, we solved the algebraic constraints to theordinary differential equations describing the economicproduction process resulting in the following equations: X c =( b c K β c c C γ ) − α , X d = ( b d K β d d ) − α ,X R = (cid:18) − b R e G G (cid:19) − α , (28a) w = αL α − ( X c + X d X R ) − α , (28b) r c = β c K c X c L α ( X c + X d X R ) − α , (28c) r d = β d K d X d X R L α ( X c + X d X R ) − α , (28d) R = b d e K β d d L α (cid:18) X d X R X c + X d X R (cid:19) α , (28e)˙ G = − R, (28f)˙ K ( i ) c = sδ o i ,c ( r c K ( i ) c + r d K ( i ) d + wL ( i ) ) − κK ( i ) c , (28g)˙ K ( i ) d = sδ o i ,d ( r c K ( i ) c + r d K ( i ) d + wL ( i ) ) − κK ( i ) d , (28h)˙ C = Y c − χC. (28i) B. Pair Approximation
To derive a macroscopic approximation of the sociallearning process described by rules 1 to 4b in Sec. II B,we make use of a Pair based proxy (PBP) process thatis derived via pair approximation from the adaptive net-work process. This proxy process is not equivalent butsufficiently close to the microscopic process approximat-ing it in terms of aggregated quantities by making certainassumptions about the properties of their microscopicstructure. The aggregated quantities of interest are: thenumber of households investing in clean capital N ( c ) , thenumber of households investing in dirty capital N ( d ) , thenumber of links between agents of the same group [ cc ]and [ dd ] as well as between the two groups [ cd ]. Sincethe total number of households N and links M are fixed,these five variables reduce to three degrees of freedom,which we parameterize as follows:¯ X = N ( c ) − N ( d ) , ¯ Y = [ cc ] − [ dd ] , ¯ Z = [ cd ] . (29)These three degrees of freedom span the reduced statespace of the social process ¯S = ( ¯ X, ¯ Y , ¯ Z ) T . The invest-ment decision making process can then be described interms of jump lengths ∆ ¯S j and jump rates W ( ¯S , ¯S +∆ ¯S j )in this state space for the different events j in the set Ω ofall possible events. Their derivation is illustrated by theexample of a clean household imitating a dirty household:The approximate rate of this event is given by W c → d = Nτ (1 − ε )(1 − ϕ ) N ( c ) N [ cd ][ cd ] + 2[ cc ] p cd . (30)In some more detail this results from • N/τ the rate of social update events, i.e., the rateof events per household times the number of house-holds, • (1 − ε ) the probability of the event not being a noiseevent, • (1 − ϕ ) the probability of imitation events (versusnetwork adaptation events), • N ( c ) /N the probability of each active household toinvest in clean capital, • [ cd ] / (2[ cc ]+[ cd ]) the approximate probability of in-teraction with a household investing in dirty capi-tal. Here, we approximate the distribution of dirtyneighbors among clean households with its first mo-ment i.e., we act as if links between clean anddirty households were evenly distributed among allhouseholds. • p cd is the expected value of the probability ofeach active household imitating its randomly cho-sen neighbor depending on the difference in con-sumption between households investing in cleanand dirty capital as given in equation (14). Theexpression is derived in detail as part of the mo-ment closure in subsection III C.The corresponding change in the state space variables isa little more tricky. Since the event is a clean householdimitating a dirty household, we already know about oneof the neighbors of the household. As laid out in detailby e.g. [38], the state of the remaining neighbors in thefull model is determined by the frequency of higher or-der network motifs, e.g., [dcd] and [dcc]. The frequency of these higher order motifs is approximated by the ex-pected value of the states of additional neighbors as fol-lows: Summing over the excess degree of the node q c bydrawing k c − p ( d ) ,or clean, p ( c ) , reads: p ( c ) = 2[ cc ]2[ cc ] + [ cd ] ; p ( d ) = [ cd ]2[ cc ] + [ cd ] . (31)This results in an expected number of n ( c ) additionalclean neighbors and n ( d ) additional dirty neighbors: n ( c ) = (1 − /k ( c ) ) 2[ cc ] N ( c ) , n ( d ) = (1 − /k ( c ) ) [ cd ] N ( c ) , (32)where k ( c ) is the mean degree, e.g., the mean number ofneighbors of a clean household in the network. With theresults from (32) the changes in the expected values ofthe state space variables can be approximated as follows:∆ N ( c ) = − , ∆ N ( d ) = 1 , ∆[ cc ] ≈ (cid:18) − k ( c ) (cid:19) cc ] N ( c ) , ∆[ dd ] ≈ (cid:18) − k ( c ) (cid:19) [ cd ] N ( c ) , ∆[ cd ] ≈ − (cid:18) − k ( c ) (cid:19) cc ] − [ cd ] N ( c ) , and, summing up, the change in the state vector is ap-proximately given by:∆ ¯S c → d ≈ − − k ( c ) − (cid:0) − k ( c ) (cid:1) cc ] − [ cd ] N ( c ) . (33)In terms of the jump lengths ∆ ¯S and the rates W , thedynamics of the PBP can be written as a master equationfor the probability distribution P on the state space of ¯S : ∂P ( ¯S , t ) ∂t = (cid:88) j ∈ Ω P ( ¯S − ∆ ¯S j , t ) W ( ¯S − ∆ ¯S j , ¯S ) − P ( ¯S , t ) W ( ¯S , ¯S + ∆ ¯S j ) . (34) C. Moment Closure
To describe the capital structure in the model thatconsists of 2 N equations of type (15) and (16), we use0the cohort of N ( c ) households investing in clean and thecohort of N ( d ) households investing in dirty capital andlook at the aggregates of their respective capital holdings:¯ K ( k ) l = N (cid:88) i δ o i k K ( i ) l . (35)Here, the upper index in ¯ K ( k ) l indicates the shared invest-ment decision of the cohort of households as opposed tothe index of the individual household before. The lowerindex still denotes the capital type. δ o i k is the KroneckerDelta.Later, we use the fact that in the limit of N → ∞ theseaggregates should converge to their expected values, e.g.,the first moments of their distribution with probabilityone. The time derivative of the aggregates defined in (35)is given by the deterministic process of capital accumu-lation (28g) and (28h) as well as terms resulting fromthe stochastic process of agents switching their savingdecisions.˙¯ K ( c ) c =˙¯ K ( c ) d =˙¯ K ( d ) c =˙¯ K ( d ) d = ( sr c − α ) ¯ K ( c ) c + sr d ¯ K ( c ) d + sw ¯ L − α ¯ K ( c ) d − α ¯ K ( d ) c sr c ¯ K ( d ) c + ( sr d − α ) ¯ K ( d ) d + sw ¯ L (cid:124) (cid:123)(cid:122) (cid:125) D ( i ) l +switching terms . (36)The switching terms for ¯ K ( c ) c result from agents chang-ing their saving decision, thereby moving their capitalendowments from the aggregate capital of the cohort ofclean investors to the aggregate of the cohort of dirty in-vestors and vice versa. We assume that each householdswitching to the other cohort is endowed with the meancapital of the cohort and that their capital endowmentis independent of the probability of switching such thatwe can describe the switching terms as a product of bothfactors. Then, we can write down the changes in cap-ital stocks explicitly including the switching terms as asimple stochastic differential equation:d ¯ K ( k ) l = D ( k ) l d t + ¯ K ( j ) l N ( j ) d N j → k − ¯ K ( k ) l N ( k ) d N k → j , (cid:124) (cid:123)(cid:122) (cid:125) switching terms (37)where the first term of the right-hand side refers to thechange in aggregates without switching, as given by theequations of capital accumulation (36) and the followingterms denote the influx and outflux of capital from theaggregate due to households changing their savings de-cisions. d N j → k denotes the stochastic process of house-holds switching from one opinion to another according tothe rules outlined in II B. In line with the pair approxi-mation described in III B we approximate them asd N j → k = (cid:88) l ∈ Ω j → k W l d t (38) where Ω j → k denotes the set of all events that result in ahousehold changing from cohort j to cohort k and W l isthe rate of the respective event analogously to (30).The imitation probability p cd in Eq. (30) is approx-imated as the expected value of a linearized version ofEq. (14) when drawing a pair of neighboring households i , j as specified. More precisely, we perform a Taylorexpansion of Eq. (14) in terms of the consumption of thetwo interacting households F ( c ) and F ( d ) around somefixed values F ( c ) ∗ and F ( d ) ∗ up to linear order. To main-tain the symmetry of the imitation probabilities with re-spect to the household incomes, we change variables to∆ F = F ( c ) − F ( d ) and F = F ( c ) + F ( d ) and expandaround ∆ F = 0 , F = F , where F is yet to be fixed to avalue. In linear order this results in: p cd = 12 − a F ∆ F, (39) p dc = 12 + a F ∆ F. (40)To make the approximation work in the biggest partof the systems state space, we set the reference point F to be the middle of the sum of the estimated upperand lower bounds for the attainable income of householdsinvesting in the clean, resp. dirty sector. The minimumattainable income is assumed to be zero. The maximumattainable income for a household investing in the cleansector is assumed to be reached in equilibrium given allother households also invest in the clean sector, e.g., wecalculate F ( c ) ∗ as half of an average household incomeat the steady state of ˙ K c = sb c L α K β c c C γ − δK c and˙ C = b c L α K β c c C γ − δC : C ∗ = (cid:18) b c L α s β c δ (cid:19) − βc − γ , K ∗ c = (cid:18) b c L α s − γ δ (cid:19) − βc − γ . (41)Equivalently, we calculate F ( d ) ∗ as half of an aver-age household income at the steady state of ˙ K d = s (cid:0) − b R e (cid:1) b d K β d d P α − δK d : K ∗ d = (cid:18) sb d L α δ (cid:18) − b R e (cid:19)(cid:19) (cid:16) − βd (cid:17) . (42)With these results, using the fact that we set β c = β d = α = 1 /
2, the reference point F is F = 12 (cid:16) F ( c ) ∗ + F ( d ) ∗ (cid:17) = 1 − s N ( r ∗ c K ∗ c + wL + r ∗ d K ∗ d + wL ) (43)= 1 − s N (cid:32)(cid:18) sb c L α δ β c + γ (cid:19) − βc − γ + sδ (cid:18)(cid:18) − b R e (cid:19) b d L α (cid:19) (cid:33) , (44)where r ∗ c and r ∗ d in (43) are the capital return rates (9)and (10) in the respective equilibria (41) and (42).1Given this linear approximation of the imitation prob-abilities, we approximate the consumption F c and F d ofthe randomly selected households i and j as the house-hold consumption of the average household investing inclean and dirty capital using the aggregated variablesas introduced in (35). In the large system limit, this isequivalent to taking the expected value over all house-holds in the respective cohorts: p cd = 12 − a F (cid:16) r c (cid:16) ¯ K ( c ) c − ¯ K ( d ) c (cid:17) + r d (cid:16) ¯ K ( c ) d − ¯ K ( d ) d (cid:17) + w LN (cid:16) N ( c ) − N ( d ) (cid:17)(cid:19) , (45) p dc = 12 + a F (cid:16) r c (cid:16) ¯ K ( c ) c − ¯ K ( d ) c (cid:17) + r d (cid:16) ¯ K ( c ) d − ¯ K ( d ) d (cid:17) + w LN (cid:16) N ( c ) − N ( d ) (cid:17)(cid:19) . (46)With this approximation, we have now reached an ap-proximate description of the microscopic dynamics interms of stochastic differential equations for the aggre-gate variables. D. Large System Limit
The description of the model in terms of equations(28f), (28i) (34) and (36) poses a significant reductionof complexity, yet it is still a description in terms of astochastic process rather than in terms of ordinary dif-ferential equations, as typically used in macroeconomicmodels. To further reduce it to ordinary differential equa-tions, we do an expansion in terms of system size, whichin our case is given by the number of households N .Therefore, following Van Kampen [94, p. 244], we in-troduce the rescaled variables x = XN , y = YM , z = ZM , k = 2
MN . (47)and expand the master equation (34) that describes thesocial learning process in terms of a small parameter N − . In the leading order, the time development of therescaled state vector s = ( x, y, z ) is given byddt s = α , ( s ) , (48)where α , is the first jump moment of W . In terms ofthe rescaled variables s , α , is given by α , ( s ) = (cid:90) ∆ s W ( s, ∆ s )d∆ s , (49)which in the case of discrete jumps in state space simpli-fies to ddt s = (cid:88) j ∈ Ω ∆ s j W j , (50) where Ω is the set of all possible (discrete) events in theopinion formation process.As for the economic processes, we keep the aggregatedquantities ( ¯ K ji , C, G ) fixed and formally go to a contin-uum of infinitesimally small households. As people andalso households for that matter are finite entities, a con-tinuum of households makes no sense. But practically,this can be understood as an interpretation of the hetero-geneous households as a weighted sample of a very largepopulation of heterogeneous individuals and increasingthe sample size up until the point where a continuum ofhouseholds is a sufficiently good approximation of real-ity in terms of the model. The only element in the ap-proximation of the economic model that depends on perhousehold quantities is the imitation probability (14) orrather its approximation (39) and (40). Since we havechosen this to depend on relative differences in income,their dependence on the number of households N cancelsout and the limit of N → ∞ becomes trivial resulting inthe following deterministic approximation for the capitalendowments in sector l of households investing in sector k described in Eq. (37):˙¯ K ( k ) l = D ( k ) l + ¯ K ( j ) l N ( j ) (cid:88) l ∈ Ω j → k W l − ¯ K ( k ) l N ( k ) (cid:88) l ∈ Ω k → j W l , (51)where D ( k ) l are the capital accumulation terms as given in(36) and Ω l → k is the set of all opinion formation events,where a household changes its opinion from l to k .Together with equations (28f) and (28i) the sets ofequations specified by (50) and (51) fully describe theapproximate dynamics of the original model as specifiedin Section II. The full set of equations is given in Ap-pendix D.Our approximation reduces the full model to a set offirst order differential equations with nine degrees of free-dom. For comparison, the full model has 2 N + 2 degreesof freedom in the economic system plus the configurationspace of the social network component. The right-handsides of the set of differential equations are continuouslydifferentiable and depend on 12 parameters for the eco-nomic system and two parameters for the social networkprocess. The state space of the system is bounded be-tween − x and y and between 0 and 1 in z aswell as by 0 from below in the variables of the economicsystem ¯ K ( k ) l , G and C . As the equations are bulky, it isrecommended to use a computer algebra system to workwith them.The freedom to chose equations for economic produc-tion that are not scale-invariant critically depends on theassumption that household interaction only depends onrelative differences. For individual interaction that de-pends on absolute differences, one can show that the largesystem limit only works if the system is scale-invariant interms of aggregated quantities. Nevertheless, it would bepossible to relax both of these assumptions and to work2 Symbol Variable description[ cc ], [ dd ], [ cd ] number of links between clean anddirty households p kl probability that household investing insector l ∈ { c, d } imitates householdinvesting in sector k ∈ { c, d } W jump rates for the stochastic processapproximating the network dynamics¯ S jump lengths for the networkdynamics¯ K ( k ) l aggregate capital investments in sector l ∈ { c, d } of all households in category k ∈ { c, d } TABLE III. List of variables used in the macroscopic approx-imation with the PBP process with the results explicitly depend-ing on the number of households, which in return couldlead to interesting finite size effects.
E. Results of the Model Approximation
The results in Fig. 3 are to some extent complemen-tary to the results in Fig. 2 that we discussed in Sec. II C.Fig. 3d shows capital in both sectors belonging to house-holds that actually invest in these sectors, which is almostequivalent to the variables in fig. 2d as it makes up almostthe entirety of these capital stocks. This can be seen inFig. 3c: It shows capital of households in the sector thatthey do not currently invest in, which is approximatelyan order of magnitude smaller (note the different scale ofthe vertical axis in the figure).A comparison of the results of the approximation(dashed lines) with those of the numerical simulation ofthe ABM (solid lines) in Fig. 3 shows that the approx-imation exhibits the same qualitative features, such astrends, timing and order of magnitude of the displayedvariables, as the microscopic model.Particularly, these results show that for the given pa-rameter values the macroscopic approximation is capableof reproducing very closely the quasi-equilibrium statesbefore and after the transition from the dirty to the cleansector, as it lies within the standard error of the ensembleof ABM runs. Also, the approximation is reasonably ca-pable to reproduce the timing of and the transient statesduring the transition. This is somewhat surprising sincein other works, macro-approximations were less well ableto get the timing of transition right.In the following, we discuss the existing differences be-tween the results of the approximated model and the nu-merical simulation results.For instance, we find that the approximation estimatesthe transition from investment in the dirty sector to in-vestment in the clean sector a bit too early (best visiblein panel a). The reason for this might be the slight un- derestimation of the share of clean investing households,leading to a slight overestimation of the share of dirtycapital in the system which is also visible in panel 3c.We find a second obvious discrepancy between themicro-model and the approximation in the overestima-tion of dirty capital of clean investors ( K ( c ) d ) (paneld) during the transition phase between t ≈
150 and t ≈ f r a c t i o n o f n o d e s , e dg e s a) N c / N [ cc ]/ M [ cd ]/ M k n o w l e dg e s t o c k C ×10 b) CG c a p i t a l s t o c k K ( j ) i ×10 c) K ( c ) c K ( d ) d c a p i t a l s t o c k K ( j ) i ×10 d) K ( d ) c K ( c ) d r e s o u r c e s t o c k G ×10 FIG. 3.
Trajectories of dynamic variables from the macro approximation and from measurement in ABMsimulations.
The results from ABM simulations (solid lines) are obtained as an ensemble average from 50 runs with standarderrors indicated by gray areas. Initial conditions are given by equal shares of the N = 200 households investing in both sectorsand equal endowments in both sectors for all households. The initial acquaintance network amongst the households is an Erd˝os-Renyi random graph with mean degree k = 10. Other initial conditions are C = 0 . G = 5 × . All other parameterare given in table II. The results from the macro approximation (dashed lines of the same colors) are obtained by integration ofthe ODEs that are obtained from the large system limit with fixed per household quantities. The initial conditions are drawnfrom the same distribution as previously for the ABM simulations e.g. N c , [ cc ] and [ cd ] are calculated from an Erd˝os-Renyirandom graph with mean degree k = 10. clean investors ( K ( c ) d ) during the transition as well asthe underestimation of ( K ( c ) d ) before the transition andtherefore also estimate the timing of the transition evenmore precisely.Similarly, a higher-order motif approximation of thenetwork dynamic can describe the heterogeneity inthe local distribution of opinions in the neighborhoodof individual agents and correct for the effects of thisespecially during periods of transient non-equilibriumdynamics in the approximated model.In the previous section we derived a set of ordinary dif-ferential equations describing the stochastic dynamics ofan agent-based model in terms of aggregated variables inthe large system limit. We intend this derivation to be aprototypical example for a macroeconomic model withtrue microfoundations based on heterogeneous agents,given their microscopic interactions are of similar com-plexity. As such, it might also serve as a starting pointfor the application and development of similar models for other kinds of social dynamics. For example, an exten-sion to continuous opinions requiring a Fokker-Planck-type description would follow naturally and would grantcompatibility to a large body of models for social influ-ence [see ref. 95, pp. 988 f.]. IV. BIFURCATION ANALYSIS
The description of the model as a system of ordinarydifferential equations allows for the analytical analysis ofemergent model properties such as multi-stability, tip-ping and phase transitions. As a proof of concept appli-cation we subsequently show the results of a bifurcationanalysis.4 C P1 P2LP1LP2 0.120 0.125 0.130 0.135 0.14015002000250030003500 K c d P1 P2LP1LP20.120 0.125 0.130 0.135 0.1400.80.60.40.20.00.20.40.60.8 x P1 P2LP1LP2 0.120 0.125 0.130 0.135 0.1400.150.200.250.300.350.40 z P1 P2LP1LP2 a) b)c) d)
FIG. 4.
Bifurcation diagram:
Continuation of the stationary solution of the macroscopic approximation without resourcedepletion, i.e., with ˙ G = 0 instead of the rate R as given by Eq. (28f). Bifurcation parameter is γ , the elasticity of knowledgein the clean sector that also reflects the elasticity of learning by doing of the respective technology. The points labeled P1 andP2 are the beginning and end points of the continuation line, the points labeled LP1 and LP2 are the bifurcation points oftwo fold bifurcations. The stable unstable manifold is indicated by a dotted line, the stable manifold is indicated by solid line.Note that the intersections of the curves in the two right panels do not actually mean that the stationary manifold is not abijective function of the bifurcation parameter γ but rather a result of the projection of the multidimensional manifold ontothe two-dimensional space. A. Methods
Bifurcation theory is the analysis of qualitative changesof dynamical systems under parameter variation, for ex-ample between a regime with a unique equilibrium (fixedpoint) and a multi-stable regime. The parameter valueat which a qualitative change, for example in the sta-bility of an equilibrium, occurs is called a critical valueor bifurcation point. Bifurcations are classified accord-ing to the changes in dynamical properties of the system[96, 97]. Analytical methods have limited scope to iden-tify bifurcation points in non-linear systems. Methods like numerical continuation can handle complex systemsof ordinary differential equations like the one derived inSec. III [98]. Consequently, we use numerical continu-ation from PyDSTool [99, 100], a Python package fordynamical systems modeling and analysis [101].A common bifurcation type that appears in our modelis the fold bifurcation that is also known as saddle-nodebifurcation. This type is a local bifurcation in which astable fixed point collides with an unstable one and bothdisappear.Varying two bifurcation parameters at the same timecan result in even richer qualitative changes of the dy-5namics. A prevalent example for such a bifurcation isthe cusp geometry [97, p. ˙397]. A change of the secondbifurcation parameter in this geometry beyond a certainvalue results in the so-called cusp catastrophe: the multi-stability of the system disappears for all values of the firstbifurcation parameter. As we will show in the following,the macro-approximation of our model indeed exhibits acusp bifurcation.
B. Discussion of Results C b d FIG. 5.
Cusp Bifurcation diagram:
Stationary manifoldfrom Fig. 4 panel a for different values of the total factorproductivity on the dirty sector b d . Red dots indicate thelimit points of the one dimensional fold bifurcation separatingthe stable and the unstable parts of the stationary manifoldindicated by a solid and a dashed line respectively. For acritical value of b d ≈ . γ ≈ . γ and b d is called acusp catastrophe. In our two-sector economic model, thisresults in a lock in effect in the dirty sector, i.e., below thispoint, there is a smooth transition of production from thedirty to the clean sector and above this point production inthe dirty sector is continued even though production in theclean sector would be more efficient. A considerable advantage of the description of ourmodel in terms of ordinary differential equations (28f),(28i) (50) and (51) over agent-based modeling is the factthat it allows for the usage of established tools for bi-furcation analysis. As a proof of concept, we show someresults in Fig. 4. Here, we analyze the possible steadystates of the system with abundant fossil resources, e.g.,the possible equilibrium states of the model in the regimebefore the fossil resource becomes scarce and acts as anexternal driver on the system pushing it towards clean investment. Therefore, we set the resource depletion tozero, i.e., we keep the resource stock in Eq. (28f) constant G ( t ) ≡ G such that the resource usage cost in Eq. (6)still depends on resource use R but is not increased bydeceasing resource stock G . Thereby, we eliminate therising resource extraction cost as the constraint in (7) and(10) that eventually halts production in the dirty sector.We chose the learning rate γ as bifurcation parameteras we expect it to yield interesting results. Generally,in nonlinear dynamical systems, exponential factors areexpected to have a strong influence on dynamical proper-ties. Therefore, changing these factors is expected to leadto bifurcation behavior. Consequently, in Fig. 4 panel aand c we see that for certain learning rates γ the macro-scopic approximation exhibits a bistable regime limitedby two fold bifurcations with bifurcation points indicatedby LP1 and LP2. In this regime both low investmentin the clean sector together with high investment in thedirty sector and low knowledge as well as high invest-ment in the clean sector together with low investmentin the dirty sector and high knowledge are stable statesof the economic system. This means that in this regioneconomic outcomes are highly path dependent. Startingwith slightly different knowledge about clean technolo-gies may lead to widely differing adoption levels of thetechnology in the long run.Figure 5 shows an example of how this bifurcationstructure of the dynamical system depends on other pa-rameters. Varying the total factor productivity in thedirty sector, b d , the system undergoes a cusp bifurca-tion. Above a certain value of b d the system exhibitsbi-stability whereas below this value it does not.Clearly, this choice of bifurcation parameters is onlyone of many and other choices may very well lead tointeresting results. However, we had to limit ourselves tothis proof of concept study since an extensive analysis ofall possible combinations would be well beyond the scopeof this paper.Multi-stability of the economy would mean that poli-cies could make use of inherent dynamical properties ofthe system to reach a desired state or bring the sys-tem onto a desired pathway. For example, policy mea-sures such as regulation or taxes can help driving thesystem into another basin of attraction, i.e., a region ofthe phase-space in which trajectories approach anotherequilibrium in the long term. To do so, the system hasto cross a separatrix, the boundary between two basinsof attraction. After this boundary is crossed, the pol-icy measure can be discontinued, the system’s dynamicsguarantee that it reaches the new equilibrium. Figure 5shows that such an intervention could be complementedby an additional policy measure, lowering the total fac-tor productivity in the dirty sector, effectively reducingthe distance of the stable manifold from the separatrixand thereby presumably making the first measure lesscostly. Another possibility to take advantage of the sys-tem’s inherent dynamical structure is to use its hystere-sis, i.e., to find policy measures that change the first bi-6furcation parameter γ across a bifurcation point or tochange the second bifurcation parameter b d to move thebifurcation point past the current state of the system (ora combination of both) after which the system would fallto the other branch of the stable manifold. Afterwards,the policy can be discontinued and the system would re-main in its new state. For such considerations, tools fromdynamical systems theory and topology can be used toclassify the phase-space of the system into regions withrespect to the reachability of a desirable state [93, 102].This allows designing temporary policies that leveragethe multi-stability of the socio-economic system. V. CONCLUSION
This paper combines a set of methods to overcomeshortcomings of current approaches to base macroeco-nomic models on microfoundations. While representativeagent approaches are unable to capture dynamics thatemerge from structured and local interactions of multi-ple heterogeneous agents, computational agent-based ap-proaches have the disadvantage that they make tractablemodel analysis difficult and computationally challeng-ing. We demonstrated that a combination of approxi-mation techniques allows finding a macro description ofa multi-agent system in which heterogeneous agents in-teract locally on a complex adaptive network as well asvia aggregated quantities. In contrast to previous ana-lytic work, where the network structure was either static[36], restricted to star-like clusters [23] or approximatedby a mean-field interaction approach and hence neglected[24, 25, 29, 30, 35], we explicitly treat the structure of theadaptive complex interaction network with appropriateapproximation methods.We develop a stylized two-sector investment model, inwhich investment decisions are driven by a social imi-tation process, to showcase the three approximations:First, a pair approximation of networked interactionstakes into account the heterogeneity in interaction pat-terns. Second, a moment closure approximation makes itpossible to deal with heterogeneous attributes that char-acterize the agents. Third, the large-system limit ab-stracts from effects due to finite population size. It isonly possible to take this limit if the model has at leastone of the following properties: (i) individual interactiondepend only on relative rather than absolute quantitiessuch that the size of households can be decreased whiletaking the number of households to infinity or (ii) theeconomic production functions exhibit constant returnsto scale such that they scale linearly with the number ofhouseholds N . The resulting set of ordinary differentialequations captures the effect of local interactions at thesystem level while still allowing for analytical tractability.A comparison between a computational version of theABM and the macro-description reveals that the approx-imation works well for parameter values distinct fromspecial cases even if only accounting for first moments. Taking more moments into account would increase accu-racy but comes at the cost of higher dimensionality andcomplexity of the macroscopic dynamical system.Our model shows that social imitation dynamics addinertia to the investment decisions in the system thatcannot be captured by a representative agent approach.The imitation process results in social learning such thatagents tend to direct their investments into the moreprofitable sector over time. Because of this, the shift ofinvestments from the dirty (fossil) to the clean (renew-able) sector is driven only by economic factors, namelyincreasing exploration and extraction costs for the fossilenergy resource. Thus, we conclude that neutral imita-tion of better-performing peers is not a feasible mecha-nism to initiate a bottom-up transformation of the econ-omy. Directed imitation, for example driven by changesin social norms, and supporting policies that make dirtyproduction less profitable are needed to initiate a trans-formation towards a sustainable economy in the absenceof fossil resource shortage.Finding a system of ordinary differential equations toapproximate ABMs is useful because it makes the analy-sis of the dynamical properties of the model much easier.One promising application here is bifurcation theory, asillustrated in Sec. IV. Furthermore, it opens the possibil-ity to mathematically prove model properties such as thedependency between different parameters and variablesin the model.In the context of climate economics and policy, the pro-posed techniques are especially important because theyallow investigating the interplay of learning agents adapt-ing to new policies and effects of shifts in values andpreferences. The resulting changes in individual behav-ior and their impact on macroeconomic dynamics can bestudied in a comprehensive modeling framework. Largeshifts in investments that are required to reach the goalsof the Paris agreement are likely to profit from both,policies that rely on price signals, as well as policies thattarget individual norm change, interaction and behav-ior not unlike those researched in, e.g., the public healthcontext [86, 103, 104]. The presented techniques can helpto better understand how such behavioral interventionswould impact the macro-level dynamics of the economicsystem.On this regard, there are several promising avenues todevelop the model and approximation techniques further:For example, instead of binary opinions, the social inter-action model can use continuous variables to representgradual opinions, drawing on a variety of models of so-cial influence [see ref. 95, pp. 988 f.]. An approximationof the agent ensemble would then need a Fokker-Planck-type description rather than a master equation.Our model could be extended to explicitly include pol-icy instruments such as a carbon tax and explore its im-pact on the investment decisions of the heterogeneousagent population. Another promising modification couldinclude consumption decisions into our two-sector model.Consumption decisions are strongly influenced by social7norms and interactions [105]. Their inclusion could in-form the discussion about green consumption as a poten-tial mechanism for a bottom-up transformation towardsa more sustainable economy.Finally, the techniques proposed in this paper couldbe used to approximate other systems that interact bothlocally on a network and in an aggregate way on the sys-tem level, for example social-ecological systems or neuralnetworks. ACKNOWLEDGEMENTS
The authors declare that they do not have any conflictof interest. Jakob J. Kolb acknowledges funding by the Foundation of German Industries. Finn M¨uller-Hansenacknowledges funding by the DFG (IRTG 1740/TRP2011/50151-0). Jobst Heitzig acknowledges funding bythe Leibniz Assiciation (project DominoES). Numericalanalyses were performed on the high-performance com-puter system of the Potsdam-Institute for Climate Im-pact Research, supported by the European Regional De-velopment Fund, BMBF, and the Land Brandenburg.
Appendix A: Comparing adaptive with fully connected network
300 400 500 600 700t0.00.20.40.60.81.0 N _ c o v e r N
300 400 500 600 700t0.10.20.30.4 [ c d ] o v e r M = 0.5 = 0.8 = 0.9 FIG. 6. Comparison of microscopic model with adaptive network dynamics with microscopic model with fully connected networkfor varying rewiring rate ϕ . All other parameters are given in Tab. II. Solid lines indicate results with network adaptation,dashed lines indicate results with fully connected network. Initial network topology is a Erd˝os-Renyi random graph. We compare the dynamics of the micro model with adaptive network rewiring with the dynamics of micro modelwith a fully connected acquaintance network. The model with a fully connected acquaintance network is equivalentto a well-mixed model with pairwise interactions between all agents. The results in Fig. 6 show, that the well-mixedmodel approximates the adaptive network model for ϕ = 0 . ϕ , the fragmentationincreases in the adaptive network model, indicated by the lower fraction of links between agents with different savingsdecisions (clean and dirty) [ cd ] /M . This cannot be captured by the fully connected network model. As an economicallyobservable result, this leads to significantly slower tipping in the adaptive network model. Appendix B: Effects of the rewiring rate ϕ on model dynamics We analyze the effect of changes in the network rewiring rate ϕ on the model dynamics. The results in Fig. 7indicate that for increasing rewiring rate ϕ the model undergoes a transition from a connected network state witha considerable number of connections between agents investing in different sectors, to a fragmented network statein which such connections are effectively non-existent. This transition is especially apparent in the fraction of [ cd ]links in the network given in Fig. 7b. This fragmentation transition is well known for adaptive voter type models[39, 41, 90].8
300 400 500 600 700 800t0.00.20.40.60.8 N _ c o v e r N
300 400 500 600 700 800t0.00.10.20.30.4 [ c d ] o v e r M
300 400 500 600 700 800t0.050.100.150.200.25 r _ c
300 400 500 600 700 800t050010001500200025003000 Y _ c = 0.0= 0.5= 0.75= 0.85= 0.9= 0.95 FIG. 7. Model trajectories with varying ϕ . All other parameters are given in Tab. II. Results are ensemble averages from 200runs. Initial network topology is a Erd˝os-Renyi random graph. Appendix C: Model dynamics depending on network rewiringAppendix D: ODEs resulting from approximation
The following are the full ordinary differential equations resulting from (50) ,(51), (28f) and (28i)˙ x = − (cid:15)xτ − p cd z ( (cid:15) −
1) ( φ −
1) ( x + 1) τ ( y + 1) + p dc z ( (cid:15) −
1) ( φ −
1) ( x − τ ( y −
1) (D1)˙ y = − m ( p cd z ( (cid:15) −
1) ( φ − − p dc z ( (cid:15) −
1) ( φ −
1) + 0 . (cid:15) ( y −
1) + 0 . (cid:15) ( y + 1)) τ + ( x −
1) (0 . (cid:15)z ( x − − . (cid:15) ( x + 1) ( y + z −
1) + 0 . φz ( (cid:15) − τ ( y − x + 1) (0 . (cid:15)z ( x + 1) + 0 . (cid:15) ( x −
1) ( y − z + 1) − . φz ( (cid:15) − τ ( y + 1) (D2)˙ z = − (cid:15)m (2 z − τ − . p cd z ( (cid:15) −
1) ( φ −
1) (( x + 1) ( y + 1) − y − z + 1) ( my + m − . x − . τ ( y + 1) − . p dc z ( (cid:15) −
1) ( φ −
1) (( x −
1) ( y − − y + 2 z −
1) ( my − m − . x + 0 . τ ( y − + ( x −
1) (0 . (cid:15)z ( x − − . (cid:15) ( x + 1) ( y + z −
1) + 0 . φz ( (cid:15) − τ ( y − − ( x + 1) (0 . (cid:15)z ( x + 1) + 0 . (cid:15) ( x −
1) ( y − z + 1) − . φz ( (cid:15) − τ ( y + 1) (D3)˙ K ( c ) c = K ( c ) c ( − δ + r c s ) + K ( c ) d r d s + Lsw − . K ( c ) c ( x + 1) ( p cd z ( (cid:15) −
1) ( φ −
1) + 0 . (cid:15) ( y + 1)) τ ( y + 1)+ 0 . K ( d ) c ( x −
1) ( p dc z ( (cid:15) −
1) ( φ − − . (cid:15) ( y − τ ( y −
1) (D4)˙ K ( d ) d = K ( d ) d ( − δ + r d s ) + K ( d ) c r c s + Lsw + 0 . K ( c ) d ( x + 1) ( p cd z ( (cid:15) −
1) ( φ −
1) + 0 . (cid:15) ( y + 1)) τ ( y + 1) − . K ( d ) d ( x −
1) ( p dc z ( (cid:15) −
1) ( φ − − . (cid:15) ( y − τ ( y −
1) (D5)˙ K ( c ) d = − K ( c ) d δ − . K ( c ) d ( x + 1) ( p cd z ( (cid:15) −
1) ( φ −
1) + 0 . (cid:15) ( y + 1)) τ ( y + 1)+ 0 . K ( d ) d ( x −
1) ( p dc z ( (cid:15) −
1) ( φ − − . (cid:15) ( y − τ ( y −
1) (D6)˙ K ( d ) c = − K ( d ) c δ + 0 . K ( c ) c ( x + 1) ( p cd z ( (cid:15) −
1) ( φ −
1) + 0 . (cid:15) ( y + 1)) τ ( y + 1) − . K ( d ) c ( x −
1) ( p dc z ( (cid:15) −
1) ( φ − − . (cid:15) ( y − τ ( y −
1) (D7)(D8)0˙ G = − L π b d e R (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:17) − π (cid:16) − G b R G e R (cid:17) − π (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:17) − π (cid:16) − G b R G e R (cid:17) − π + (cid:16) C ξ b c (cid:16) K ( c ) c + K ( d ) c (cid:17) κ c (cid:17) − π π (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (D9)˙ C = − Cδ + C ξ b c L (cid:16) C ξ b c (cid:16) K ( c ) c + K ( d ) c (cid:17) κ c (cid:17) − π (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:17) − π (cid:16) − G b R G e R (cid:17) − π + (cid:16) C ξ b c (cid:16) K ( c ) c + K ( d ) c (cid:17) κ c (cid:17) − π π × (cid:18) K ( c ) c (cid:18) x (cid:19) + K ( d ) c (cid:18) − x (cid:19)(cid:19) κ c (D10)where p cd and p dc are given by Eq. (45) and (46) and r c , r d and w are given by r c = L π κ c (cid:16) C ξ b c (cid:16) K ( c ) c + K ( d ) c (cid:17) κ c (cid:17) − π (cid:18) C ξ − π b − π c (cid:16) K ( c ) c + K ( d ) c (cid:17) κc − π + (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:16) − G b R G e R (cid:17)(cid:17) − π (cid:19) − π K ( c ) c + K ( d ) c (D11) r d = L π κ d (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:16) − G b R G e R (cid:17)(cid:17) − π K ( c ) d + K ( d ) d × (cid:32) C ξ − π b − π c (cid:16) K ( c ) c + K ( d ) c (cid:17) κc − π + (cid:18) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:18) − G b R G e R (cid:19)(cid:19) − π (cid:33) − π (D12) w = L π − π (cid:32) C ξ − π b − π c (cid:16) K ( c ) c + K ( d ) c (cid:17) κc − π + (cid:16) b d (cid:16) K ( c ) d + K ( d ) d (cid:17) κ d (cid:17) − π (cid:18) − G b R G e R (cid:19) − π (cid:33) − π (D13) [1] V. Grimm and S. F. Railsback, Individual-based mod-eling and ecology , Princeton Series in Theoretical andComputational Biology (Princeton University Press,Princeton, NJ, 2005).[2] E. Bonabeau, Agent-based modeling: Methods andtechniques for simulating human systems, Proceedingsof the National Academy of Sciences , 7280 (2002).[3] M. W. Macy and R. Willer, From Factors to Actors:Computational Sociology and Agent-Based Modeling,Annual Review of Sociology , 143 (2002).[4] L. Tesfatsion, Agent-Based Computational Economics:A Constructive Approach to Economic Theory, in Hand-book of Computational Economics Vol. 2 , edited byL. Tesfatsion and K. L. Judd (North Holland, Amster-dam, 2006) Chap. 16, pp. 831–880.[5] L. Hamill and N. Gilbert,
Agent-Based Modelling inEconomics (Wiley, Chichester, UK, 2016).[6] J. M. Epstein, Agent-based computational models andgenerative social science, Complexity , 41 (1999).[7] T. Gross and B. Blasius, Adaptive coevolutionary net-works: a review., Journal of the Royal Society, Interface/ the Royal Society , 259 (2008).[8] P. Holme and M. E. J. Newman, Nonequilibrium phasetransition in the coevolution of networks and opinions,Physical Review E , 056108 (2006).[9] L. Bargigli and G. Tedeschi, Interaction in agent-based economics: A survey on the network approach, Phys-ica A: Statistical Mechanics and its Applications ,1 (2014).[10] M. Granovetter, The Impact of Social Structure on Eco-nomic Outcomes, Journal ofEconomic Perspectives ,33 (2005).[11] D. Delli Gatti, E. Gaffeo, M. Gallegati, G. Giulioni, andA. Palestrini, Emergent Macroeconomics. An Agent-Based Approach to Business Fluctuations , New Eco-nomic Windows (Springer, Milan, 2008).[12] Approaches to represent heterogeneous agents in DSGEmodels have been used to counter this criticism andadd more realism regarding the distribution of agentattributes [see for example the review by 106]. Particu-larly, because the representative agent approach cannotaccount for interactions within a heterogeneous group,models using this approach do not allow for the rep-resentation of emergent phenomena Kirman [107]. Buttheir solution require complex numerical methods andcannot integrate local interactions between agents.[13] A. Kirman, Is it rational to have rational expectations?,Mind and Society , 29 (2014).[14] G. W. Evans and G. Ramey, Adaptive expectations, un-derparameterization and the Lucas critique, Journal ofMonetary Economics , 249 (2006).[15] We use here a weak notion of emergence, which al- lows explaining macro-phenomena on the basis of micro-interactions of the systems constituents that differ fromthe explained macro-phenomena. This is opposed tostrong emergence, that embraces the irreducibility ofmacro-phenomena to lower-level dynamics. For a dis-cussion see Bedau [108].[16] R. Leombruni and M. Richiardi, Why are economistssceptical about agent-based simulations?, Physica A:Statistical Mechanics and its Applications , 103(2005).[17] V. Grimm, U. Berger, F. Bastiansen, S. Eliassen,V. Ginot, J. Giske, J. Goss-Custard, T. Grand, S. K.Heinz, G. Huse, A. Huth, J. U. Jepsen, C. Jørgensen,W. M. Mooij, B. M¨uller, G. Pe’er, C. Piou, S. F. Rails-back, A. M. Robbins, M. M. Robbins, E. Rossmanith,N. R¨uger, E. Strand, S. Souissi, R. A. Stillman, R. Vabø,U. Visser, and D. L. DeAngelis, A standard protocolfor describing individual-based and agent-based models,Ecological Modelling , 115 (2006).[18] J.-S. Lee, T. Filatova, A. Ligmann-Zielinska,B. Hassani-Mahmooei, F. Stonedahl, I. Lorscheid,A. Voinov, G. Polhill, Z. Sun, and D. C. Parker,The Complexities of Agent-Based Modeling OutputAnalysis, Journal of Artifical Societies and SocialSimulations , 4 (2015).[19] R. N. Mantegna and H. E. Stanley, Introduction toeconophysics: correlations and complexity in finance (Cambridge university press, 1999).[20] C. Castellano, S. Fortunato, and V. Loreto, Statisticalphysics of social dynamics, Reviews of Modern Physics , 591 (2009).[21] A. D. Martino and M. Marsili, Statistical mechanicsof socio-economic systems with heterogeneous agents,Journal of Physics A: Mathematical and General ,R465 (2006).[22] D. Acemoglu, A. Ozdaglar, and A. Tahbaz-Salehi, Net-works, Shocks, and Systemic Risk (2015).[23] C. Di Guilmi, M. Gallegati, S. Landini, and J. E.Stiglitz, Towards an Analytical Solution for AgentBased Models: an Application to a Credit Net-work Economy, in Complexity and Institutions: Mar-ket Norms and Corporations , edited by A. Masahiko,B. Kenneth, D. Simon, and G. Herbert (PalgraveMacmillan, New York, 2012) Chap. 3, pp. 63–80.[24] M. Aoki,
New Approaches to Macroeconomic Modeling (Cambridge University Press, Cambridge, UK, 1996).[25] M. Aoki and H. Yoshikawa,
Reconstructing Macroe-conomics: A Perspective from Statistical Physics andCombinatorial Stochastic Processes (Cambridge Univer-sity Press, Cambridge, UK, 2006).[26] D. Delli Gatti, M. Gallegati, and A. Kirman, eds.,
In-teraction and Market Structure , Lecture Notes in Eco-nomics and Mathematical Systems (Springer, Berlin,2000).[27] S. Gualdi, M. Tarzia, F. Zamponi, and J. P. Bouchaud,Tipping points in macroeconomic agent-based models,Journal of Economic Dynamics and Control , 29(2015).[28] S. Gualdi, M. Tarzia, F. Zamponi, and J.-P. Bouchaud,Monetary Policy and Dark Corners in a stylized Agent-Based Model, Journal of Economic Interaction and Co-ordination , 507 (2017).[29] C. Di Guilmi, M. Gallegati, and S. Landini, Economicdynamics with financial fragility and mean-field inter- action: A model, Physica A: Statistical Mechanics andits Applications , 3852 (2008).[30] C. Chiarella and C. Di Guilmi, The financial instabilityhypothesis: A stochastic microfoundation framework,Journal of Economic Dynamics and Control , 1151(2011).[31] S. Landini and M. Gallegati, Heterogeneity, interactionand emergence: effects of composition, InternationalJournal of Computational Economics and Econometrics , 339 (2014).[32] J.-P. Bouchaud, Crises and Collective Socio-EconomicPhenomena: Simple Models and Challenges, Journal ofStatistical Physics , 567 (2013).[33] D. Fiaschi and M. Marsili, Economic interactionsand the distribution of wealth, in Econophysics andEconomics of Games,Social Choices and Quantita-tive Techniques , edited by B. Basu, S. Chakravarty,B. Chakrabarti, and K. Gangopadhyay (Springer, Mi-lano, 2010) pp. 61–70.[34] N. E. Friedkin and E. C. Johnsen,
Social Influence Net-work Theory (Cambridge University Press, New York,2011).[35] S. Alfarano, T. Lux, and F. Wagner, Time variationof higher moments in a financial market with heteroge-neous agents: An analytical approach, Journal of Eco-nomic Dynamics and Control , 101 (2008).[36] T. Lux, A model of the topology of the bank - firm creditnetwork and its role as channel of contagion, Journal ofEconomic Dynamics and Control , 36 (2016).[37] T. Gross and H. Sayama, Adaptive networks, in Adap-tive networks (Springer, 2009) pp. 1–8.[38] A.-L. Do and T. Gross, Contact processes and momentclosure on adaptive networks, in
Adaptive Networks:Theory, Models and Applications , edited by T. Grossand H. Sayama (Springer and NECSI, Cambridge, Mas-sachusetts, 2009) Chap. 9, pp. 191–208.[39] G. Demirel, F. Vazquez, G. A. B¨ohme, and T. Gross,Moment-closure approximations for discrete adaptivenetworks, Physica D: Nonlinear Phenomena , 68(2014).[40] M. Wiedermann, J. F. Donges, J. Heitzig, W. Lucht,and J. Kurths, Macroscopic description of complexadaptive networks co-evolving with dynamic nodestates, Physical Review E , 1 (2015).[41] B. Min and M. S. Miguel, Fragmentation transitions ina coevolving nonlinear voter model, Scientific Reports , 1 (2017).[42] C. Kuehn, Moment Closure - A Brief Review, in Con-trol of Self-Organizing Nonlinear Systems. Understand-ing Complex Systems , edited by E. Sch¨oll, S. Klapp, andP. H¨ovel (Springer, Cham, 2016) Chap. 13, pp. 253–271.[43] IPCC,
Climate Change 2014: Mitigation of ClimateChange: Contribution of Working Group III to the FifthAssessment Report of the Intergovernmental Panel on (Cambridge University Press, Cambridge, 2014).[44] A. Ansar, B. Caldecot, and J. Tibury, Stranded assetsand the fossil fuel divestment campaign: what does di-vestment mean for the valuation of fossil fuel assets?,SSEE , 1 (2013).[45] L. Mattauch and C. Hepburn, Climate Policy WhenPreferences Are Endogenous and Sometimes They Are,Midwest Studies In Philosophy , 76 (2016).[46] L. Mattauch, C. Hepburn, and N. Stern, Pigou pushespreferences: decarbonization and endogenous values (2018).[47] E. Gsottbauer and J. C. J. M. V. D. Bergh, Environ-mental Policy Theory Given Bounded Rationality andOther-regarding Preferences, Environmental and Re-source Economics , 263 (2011).[48] H. Hong and M. Kacperczyk, The price of sin: Theeffects of social norms on markets, Journal of FinancialEconomics , 15 (2009).[49] G. Williams, Some Determinants of the Socially Re-sponsible Investment Decision: A Cross-Country Study,Journal of Behavioral Finance , 43 (2007).[50] V. Griskevicius, R. B. Cialdini, and N. J. Goldstein, So-cial norms: an underestimated and underemployed leverfor managing climate change, International Journal ofSustainability Communication , 5 (2008).[51] T. Masson and I. Fritsche, Adherence to climate change-related ingroup norms: Do dimensions of group identifi-cation matter?, European Journal of Social Psychology , 455 (2014).[52] P. C. Stern, Contributions of Psychology to LimitingClimate Change, American Psychologist , 303 (2011).[53] A. Rabinovich, T. A. Morton, and C. C. Duke, Collec-tive Self and Individual Choice: The Role of Social Com-parisons in Promoting Public Engagement with ClimateChange, in Engaging the public with climate change: Be-haviour change and communication , edited by L. Whit-marsh, S. O’Neill, and I. Lorenzoni (Earthscan, Oxon,UK, and New York, 2011) Chap. 4, pp. 66–83.[54] B. K. Nyborg, J. M. Anderies, A. Dannenberg, T. Lin-dahl, C. Schill, M. Schl¨uter, W. N. Adger, K. J. Arrow,S. Barrett, S. Carpenter, F. Stuart, C. Iii, A.-s. Cr´epin,G. Daily, P. Ehrlich, C. Folke, W. Jager, N. Kautsky,S. A. Levin, O. J. Madsen, S. Polasky, M. Scheffer, E. U.Weber, J. Wilen, A. Xepapadeas, and A. D. Zeeuw, So-cial norms as solutions, Science , 42 (2016).[55] A. Bandura, Social learning theory, in
Social learningtheory , Vol. 1, edited by A. Bandura and R. H. Walters(Prentice-hall, Englewood Cliffs, NJ, 1977) pp. 1–46.[56] N. E. Friedkin, Norm formation in social influence net-works, Social Networks , 167 (2001).[57] D. Centola, J. C. Gonza, and M. S. Miguel, Ho-mophily, Cultural Drift, and the Co-Evolution of Cul-tural Groups, Journal of Conflict Resolution , 905(2007).[58] D. Kimura and Y. Hayakawa, Coevolutionary networkswith homophily and heterophily, Physical Review E ,016103 (2008).[59] International Monetary Fund, World economic and fi-nancial surveys , Tech. Rep. (2011).[60] R. H¨ossinger, C. Link, A. Sonntag, J. Stark,R. H¨osslinger, C. Link, A. Sonntag, and J. Stark, Esti-mating the price elasticity of fuel demand with statedpreferences derived from a situational approach, Trans-portation Research Part A: Policy and Practice ,154 (2017).[61] X. Labandeira, J. M. Labeaga, and X. L´opez-otero, Ameta-analysis on the price elasticity of energy demand,Energy Policy , 549 (2017).[62] H. E. Daly, Georgescu-Roegen versus Solow / Stiglitz,Ecological Economics , 261 (1997).[63] N. Georgescu-Roegen, Energy and economic myths,Southern Economic Journal , 347 (1975).[64] N. Georgescu-Roegen, Comments on the papers by Dalyand Stiglitz, in Scarcity and growth reconsidered , edited by V. K. Smith (Resources for the Future, New York,1979) pp. 95–105.[65] R. U. Ayres, H. Turton, and T. Casten, Energy effi-ciency, sustainability and economic growth, Energy ,634 (2007).[66] R. U. Ayres, J. C. J. M. van den Bergh, D. Linden-berger, and B. S. Warr, The Underestimated Contribu-tion of Energy to Economic Growth, Structural Changeand Economic Dynamics , 79 (2013).[67] P. Mulder and H. L. F. D. Groot, Structural change andconvergence of energy intensity across OECD countries,1970 2005, Energy Economics , 1910 (2012).[68] L. Argote and D. N. Epple, Learning curves in manu-facturing, Science , 920 (1990).[69] T. P. Wright, Factors affecting the cost of airplanes,Journal of the aeronautical sciences , 122 (1936).[70] B. Nagy, J. D. Farmer, Q. M. Bui, and J. E. Trancik,Statistical Basis for Predicting Technological Progress,PLoS ONE , 1 (2013).[71] S. Kahouli-Brahmi, Technological learning in energy-environment-economy modelling: A survey, Energy Pol-icy , 138 (2008).[72] P. Dasgupta and G. Heal, The Optimal Depletion of Ex-haustible Resources, The Review of Economic Studies,Ltd., Oxford University Press , 3 (1974).[73] R. Perman, Y. Ma, J. McGilvray, and M. Common, Nat-ural Resource and Environmental Economics , 3rd ed.(Pearson Education, 2003).[74] H. A. Simon, Theories of bounded rationality, in
De-cision and Organization , edited by C. B. McGuire andR. Radner (North Holland, 1972) Chap. 8, pp. 161–176.[75] H. A. Simon,
Models of bounded rationality: Empiricallygrounded economic reason , Vol. 3 (MIT press, 1982).[76] G. Gigerenzer and R. Selten,
Bounded rationality: Theadaptive toolbox (MIT Press, 2002).[77] A. Traulsen, D. Semmann, R. D. Sommerfeld, H.-J.Krambeck, and M. Milinski, Human strategy updat-ing in evolutionary games., Proceedings of the NationalAcademy of Sciences , 2962 (2010).[78] D. Barkoczi and M. Galesic, Social learning strategiesmodify the effect of network structure on group perfor-mance, Nature Communications , 1 (2016).[79] D. R. Fisher, J. Waggle, and P. Leifeld, Where DoesPolitical Polarization Come From? Locating Polariza-tion Within the U.S. Climate Change Debate, AmericanBehavioral Scientist , 70 (2013).[80] J. Farrell, Corporate funding and ideological polariza-tion about climate change, Proceedings of the NationalAcademy of Sciences , 92 (2016).[81] R. E. Dunlap, A. M. McCright, and J. H. Yarosh, ThePolitical Divide on Climate Change: Partisan Polariza-tion Widens in the U.S., Environment: Science and Pol-icy for Sustainable Development , 4 (2016).[82] A. M. McCright and R. E. Dunlap, THE POLITI-CIZATION OF CLIMATE CHANGE AND POLAR-IZATION IN THE AMERICAN PUBLIC’S VIEWS OFGLOBAL WARMING, 2001 2010, The SociologicalQuarterly , 155 (2011).[83] P. S. Hart and E. C. Nisbet, Boomerang Effects in Sci-ence Communication: How Motivated Reasoning andIdentity Cues Amplify Opinion Polarization About Cli-mate Mitigation Policies, Communication Research ,701 (2012).[84] H. T. Williams, J. R. McMurray, T. Kurz, and F. Hugo Lambert, Network analysis reveals open forums andecho chambers in social media discussions of climatechange, Glob. Environ. Chang. , 126 (2015).[85] M. McPherson, L. Smith-Lovin, and J. M. Cook, Birdsof a Feather: Homophily in Social Networks, AnnualReview of Sociology , 415 (2001).[86] D. Centola, An experimental study of homophily in theadoption of health behavior, Science , 1269 (2011).[87] Y. M. Asano, J. J. Kolb, J. Heitzig, and J. D. Farmer,Emergent inequality and endogenous dynamics in a sim-ple behavioral macroeconomic model, PNAS - In Re-view (2019).[88] J. J. Kolb, github.com/jakobkolb/pydivest (2018).[89] T. Gross, C. D’Lima, and B. Blasius, Epidemic Dynam-ics on an Adaptive Network, Phys. Rev. Lett. , 208701(2006).[90] G. A. B¨ohme and T. Gross, Analytical calculation offragmentation transitions in adaptive networks, Phys.Rev. E - Stat. Nonlinear, Soft Matter Phys. , 4 (2011),arXiv:1012.1213.[91] T. Rogers, W. Clifford-Brown, C. Mills, and T. Galla,Stochastic oscillations of adaptive networks: applicationto epidemic modelling, Journal of Statistical Mechanics:Theory and Experiment , P08018 (2012).[92] T. Rogers and T. Gross, Consensus time and conformityin the adaptive voter model, Phys. Rev. E - Stat. Non-linear, Soft Matter Phys. , 1 (2013), arXiv:1304.4742.[93] J. Nitzbon, J. Heitzig, and U. Parlitz, Sustainability,collapse and oscillations of global climate, populationand economy in a simple World-Earth model, Environ-mental Research Letters , 10.1088/1748-9326/aa7581(2017).[94] N. G. Van Kampen, Stochastic Processes in Physics andChemistry , 2nd ed. (North-Holland Personal Library,1992).[95] F. M¨uller-Hansen, M. Schl¨uter, M. M¨as, J. F. Donges,J. J. Kolb, K. Thonicke, and J. Heitzig, Towards repre-senting human behavior and decision making in Earthsystem models - an overview of techniques and ap-proaches, Earth System Dynamics , 977 (2017).[96] S. H. Strogatz, Book (Perseus Books, Reading, MA,1994).[97] Y. A. Kuznetsov,
Elements of Applied Bifurcation The- ory , 2nd ed., Vol. 112 (Springer Science & Business Me-dia, New York, Berlin, Heidelberg, 2013).[98] E. L. Allgower and K. Georg,