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THE AGENT-BASED APPROACH TO POST KEYNESIAN MACRO-MODELING

Corrado Di Guilmi

Economics Discipline Group, University of Technology Sydney Centre for Applied Macroeconomic Analysis Australian National University

1. Introduction

In recent years, a growing body of literature has stemmed from the cross-fertilization of agent-based (AB) modeling and Post Keynesian (PK) macroeconomics. A number of scholars (especially young researchers within the AB modeling community) have started embodying elements of PK theory in their models, in particular the consistency of stocks and flows and the endogeneity of money. At the same time, and perhaps as a consequence of this, junior researchers as well as established scholars in the heterodox field have started looking at AB modeling as a possible alternative to the standard aggregative PK modeling approach. The attention that the mainstream has been forced to give to some traditionally PK research topics have reinforced this trend. Obvious examples are the interaction between the financial sector and the real economy after the Great Recession and inequality after the publication of Piketty's book.

The review is preceded by some considerations about the reasons that have led to the cross-fertilization between the two fields. In particular, the paper highlights the complementarities and the common pillars from an epistemological perspective. In fact, although AB modeling is, in its essence, an agnostic modeling strategy developed outside the realm of social sciences, it is nevertheless peculiarly adept at integrating PK macro-modeling with formalized microeconomic foundations. Moreover, AB modeling can provide useful technical development of PK modeling and, by exposing young researchers and postgraduate students to the PK literature, can help establish and popularize PK economics as a sound alternative to neoclassical economics. AB modeling also benefits from this cross-fertilization: PK economics can represent a disciplining device for the modeler by narrowing down the degrees of freedom, supporting the construction of internally consistent models.

Within this stream of literature, models can be categorized into four different types: AB models that formalize Minsky's Financial Instability Hypothesis; evolutionary models with PK features; structural neo-Kaleckian models with AB features; and stock-flow consistent (SFC) AB models. Particular attention is devoted to SFC-AB models, given their growing popularity. The papers have been selected not so much on the basis of their explicit reference to a PK tradition (as for example, papers that present neo-Kaleckian AB models), but with the aim of linking different contributions that have progressively narrowed the distance between the two camps and led to the formulations of original frameworks to combine the two approaches.

The defining characteristics of the different models are introduced to highlight their particular solutions to the main issues that the construction of a PK model in a bottom-up fashion presents. Three types of obstacles can be identified. The first issue concerns the reformulation of behavioral rules, originally intended for the whole economy or for sectors, at the single agent level. The second issue, related to the first, is the computation of aggregate quantities starting from individual variables (for example, the multiplier) or the allocation and matching in markets starting from aggregate quantities (for example, the allocation of aggregate demand among suppliers). The third set of issues concerns two main critical points in SFC-AB modeling: first, how to model agents' demography (for example, what happens to the assets of bankrupted firms and which is the source of new entrants' initial endowments) and, second, how to ensure that the decentralized interactions respect the accounting consistency at every stage of computer simulations.

The remainder of the paper is organized as follows. Section 2 investigates the reasons of the cross-fertilization of the two schools and the possible mutual benefits. Section 3 surveys the AB models that attempted to formalize Minsky's Financial Instability Hypothesis. Section 4 presents the main AB models that can be included within the structuralist neo-Kaleckian and evolutionary traditions while Section 5 is devoted to the SFC-AB models. Finally, Section 6 offers some concluding remarks.

2. Background

This section discusses some possible mutual benefits coming from the cross-fertilization of the two approaches and outlines the compatibility of PK theory and AB modeling in an epistemological and historical perspective.

2.1 PK Economics and AB Modeling

For the reader unfamiliar with PK economics or AB modeling (or both), let us introduce some preliminary definitions and concepts.

PK economics is a heterodox school rooted mostly in the work of John Maynard Keynes and Michal Kalecki. The term “Post Keynesian” was introduced by Eichner and Kregel (1975) to define an alternative paradigm to monetarism rooted in the Keynes–Kalecki tradition. The core PK economics is summarized by King (2015) in the six core propositions identified by Thirlwall (1993): unemployment is determined in the goods market and cannot be reduced to a microeconomic representation of the job market; involuntary unemployment exists and is due to deficiencies in effective demand; investment determines savings and not vice versa; money is not neutral; the classical quantity theory of money is misleading; and given fundamental uncertainty, investment is determined by “animal spirits” and not by the solution to an optimization problem. Lavoie (2014) identifies five conceptual pillars of PK economics, which are discussed in Subsection 2.3 in order to compare them with the main features of AB modeling.

Operationally, PK theory is applied by a variety of different models, which are typically aggregative. Lavoie (2014, p. 43) identifies five different strands of PK authors: fundamentalist Keynesians, Kaleckians, Sraffians, Institutionalists, and Kaldorians. However, he recognizes that this classification is somewhat arbitrary since PK theory represents an internally consistent body of knowledge, and many PK models are a crossover among different strands. Accordingly, despite this eclecticism, we can coherently use the generic expression PK modeling.

AB modeling has been originally developed for the study of complex systems. The view of an economic system as complex is therefore central to AB modeling. A complex system can be defined as a system composed of different parts or subparts whose interaction among themselves and with the environment gives rise to emergent behaviors. Some elements of this characterization are worth stressing here. First, the central role that interaction and feedback effects play in the evolution of a complex system implies that agents can be heterogeneous and connected among themselves in multiple ways. Second, the presence of feedback effects can potentially generate nonlinear or chaotic dynamics. Third, as a consequence, the actions of single agents do not influence the system evolution in a predictable manner.

Leigh Tesfatsion defines AB modeling as the computational modeling of economic processes as open-ended dynamic systems of interacting agents.1 AB models are computationally solved by means of computer simulations and are used to represent systems with no predefined state of spaces, in which nonlinear, chaotic, or out-of-equilibrium behaviors are possible.

2.2 Microfoundations and Cross-Fertilization

Prior to discussing the theoretical overlap between the two modeling approaches, an inescapable underlying methodological question must be addressed: is microfoundation compatible with PK macro-modeling? In order to address the question, we need to eliminate a terminological ambiguity. As King (2015) and Skott (2012) imply, the term microfoundation has traditionally identified the neoclassical treatment of the micro-level analysis in macroeconomics. It usually implies an intertemporal optimization over an infinite time horizon by a representative agent endowed with perfect knowledge and perfect rationality. Abstracting from the fact that such microfoundation of macro-models is grossly flawed (see Kirman, 1992; Colander et al., 2008; Stiglitz and Gallegati, 2011, among many others), King (2015) and Skott (2012), from different perspectives, convincingly argue that this microfoundation is evidently incompatible with PK economics. While the term bottom-up might be preferable as more generic, in this paper we will use microfoundation to generically indicate the set of explicit assumptions adopted in a macroeconomic model to characterize the economic agents and underpin the underlying microeconomic behavior.

Di Guilmi et al. (2017) illustrate how the aggregation method proposed by AB modeling can overcome the simplification induced by the adoption of the representative agent. As they show, the exact aggregation of AB modeling is effective although it does not solve the issue of establishing an explicit relationship between micro- and macro-variables, in contrast with a microfoundation approach based on statistical mechanics (on this point see also Foley, 2017). Aggregative PK models circumvent the aggregation problem by using microeconomic behavioral rules to infer macroeconomic relationships (for example, production for investment functions). Such an approach is formally incorrect (Di Guilmi et al., 2017), and basically amounts to averaging out possible differences among agents, similarly to the aggregation method based on the representative agent.

According to Schoder (2017), the traditional treatment of microeconomic behavior in PK macro-models suffers from three types of inconsistencies. The first is methodological: while the macroeconomic level is presented formally, the microeconomic behavior is verbally described. This type of inconsistency may give rise to a second one: internal inconsistency, which occurs when different behavioral rules are mutually inconsistent. The third type of inconsistency is ontological: while the postulated rules are invariant to endogenous changes in the microenvironment, the model implicitly assumes them to endogenously adjust.

Further considerations could be added about the possible advantages of a more formal microfoundation for PK models (from the possibility of a wider engagement with the rest of the profession to a wider diffusion of PK ideas and tenets), but to remain within the scope of the present paper, the following discussion focuses on the compatible features of PK theory and AB modeling.

A deeper integration with PK economics can also benefit AB modeling. The fact that AB models are open-ended dynamical systems allows for a wider range of results compared to a standard general equilibrium approach, but it may also represent an issue for the identification of AB models and their application in policy analysis. More specifically Foley (2017), who has always advocated a wide adoption of AB modeling in economics, identifies a deep methodological issue with AB modeling related to their closure: the almost unlimited number of possible representations of agents' interactions, even in small systems. Given the sensitivity of results and policy indications to the model's specification, the presence of such a wide range of possible modeling choices can undermine the credibility of the model and consequently limit the usefulness of an AB representation of the economy. In his opinion, the rejection of the Walrasian type of equilibrium by AB scholars should not imply a rejection of the notion of equilibrium altogether but its replacement with sounder alternatives, such as the statistical equilibrium of complex systems in which the entropy is maximized.

In this respect, PK theory can provide a disciplining device for AB modeling from two different perspectives. First, by referring to a tradition with consolidated behavioral and structural assumptions, it can help limit the degrees of freedom for the modeler. Given its agnostic nature, an AB model can house any set of assumptions and elements, even from different traditions or modeling approaches. A too eclectic mix can undermine the internal consistency of a model and generate a range of possibly self-contradictory results. PK economics can represent a theoretical reference for constructing an internally consistent economic framework. Second, PK economics can lessen the identification problem of AB models by restricting the range of possible results and by providing a reference for an economic interpretation of the results well embedded in a consolidated literature.

While in theory every school of thought in macroeconomics could provide a reference for an internally consistent model and the interpretation of the results, the next subsection argues that PK theory significantly overlaps with the AB modeling approach.

2.3 PK Pillars and AB Modeling

Lavoie (2014) identifies five conceptual pillars of PK theory that he uses to stress the differences with the mainstream neoclassical approach. However, this theoretical synthesis can be used to show the compatibility of PK with the AB modeling approach, as shown by table 1. The first pillar identified by Lavoie is the epistemological realism of PK economics: “The objective of economics is to be able to tell a relevant story and to explain the way the economy actually works in the real world.” AB modeling, not being constrained by the analytical solvibility or by the necessity to find a stable equilibrium point, but rather aiming to reproduce realistic emergent properties at the aggregate level, is able to incorporate real-world behavioral features of economic agents. This is in stark contrast with the instrumentalism of the neoclassical approach, for which a theory must provide accurate predictions and identify the equilibrium point of the economy.

The second pillar concerns ontology. From this perspective, as Lavoie (2014, p. 17) recognizes, holism is a unifying character for PK economics and AB modeling for two reasons. First, in AB modeling, interactions among agents obey behavioral rules which constitute a complex social structure, providing a counterpart to the PK concept of institutional structure. Second, in AB modeling the emergent properties of the system are not implied by the behaviors of single agents but rather generated by their interaction: “emergent properties can be considered as macroeconomic paradoxes, or fallacies of composition” (Lavoie, 2014, p. 17).

The third pillar considers the type of rationality of economic agents. Both PK theory and AB modeling reject the hyperrationality assumption of the Neoclassical approach, preferring a procedural rationality or heuristics (see Dosi et al., 2005, for the AB model camp). From this perspective, Roos (2015) strongly argues that AB models are able to incorporate the radical uncertainty that permeates real-world decision making, as stressed by PKs, in contrast with the full knowledge and perfect rationality of Neoclassical theory, which rules out by construction radical uncertainty.

The fourth pillar concerns the focus of the analysis and the fact that in PK theory the economy is assumed to work within the production frontier and therefore exchange is less relevant than production and growth. From this perspective, it is worth stressing that AB models are open-ended systems and consequently they are not constrained by the necessity to identify internal and optimal equilibrium solutions and by the necessary assumption of decreasing returns.

The last conceptual pillar of PK economics is related to the political core of the theory, which is the impossibility of having free markets: the market is an institution and therefore created (and regulated) by a system of norms. Being a methodology, it is not possible to attach labels to AB modeling. However, while not being an embedded characteristic of AB modeling, this political presupposition can nevertheless be modeled and tested by AB models.

Table 1. Presuppositions of PK Theory and AB Model

Presuppositions PK AB modeling
Epistemology/Ontology Rationality Realism Environment-consistent rationality, satisficing agent Realism Limited rationality, heuristics
Method Economic core Holism, organicism Production, growth, abundance Complexity Open-ended evolving systems
Political core Regulated markets Agnostic

Note: Adapted from table 1.3 in Lavoie (2014).

Cogliano and Jiang (2016) argue for AB modeling as a suitable alternative for PK and in general heterodox models building on Lavoie's arguments. According to them, AB modeling represents a promising possibility for all the heterodox branches in economics. All heterodox schools tend to “share one theoretical commonality, that is, economic outcomes are, to a large extent, determined by the relation(s) between socioeconomic structure and the agents who reside in it(Cogliano and Jiang, 2016). In PK economics, this is represented by the fact that macroeconomic phenomena, such as inflation and unemployment, are the unintended consequences of the influence exerted by the capitalist structures and institutions on the individual behaviors. They argue that AB modeling is well-equipped for representing this theoretical structure for two main reasons. First, for its flexibility and the heterogeneity of agents. Second, for its evolutionary nature, which allows for continuous and bidirectional feedback between institutions and agents, both endowed with proper behavioral rules. The potential of AB modeling as a replacement of mainstream Neoclassical economics is also stressed by Harcourt and Kriesler (2013, p. 50).

Besides the general limits of AB modeling highlighted by Foley (2017), the possible drawbacks of the reformulation of PK macro-models as AB models are related to the three modeling issues identified in the introduction: the adaptation of macro-behavioral rules at the single-agent level; the need of a different closure for the model; and how to ensure the consistency of all stocks and flows. The solution to these issues and the need of using explicit functional forms for all the model's behavioral assumptions can on the one hand force the introduction of simplifications that might jeopardize the qualitative insights that are attainable in traditional aggregative models,2 and on the other hand, imply the introduction of additional assumptions whose consistency with the PK approach must be verified.

3. The Financial Instability Hypothesis in AB Models

This section surveys non-SFC-AB models that have attempted to formalize Minksy's Financial Instability Hypothesis. A Minskyan pattern for the credit cycle is a common feature of many of the models reviewed in this paper, but this section exclusively concerns itself with those models that have explicitly attempted to model the Financial Instability Hypothesis. Minskyan SFC-AB model are included in Section 5 in order to provide a comprehensive overview of that particular modeling method.

As Bucciarelli and Silvestri (2013) remark, the development of AB modeling and, in general, of computational economics has opened new perspectives for the modeling of Minsky's Financial Instability Hypothesis.

AB modeling is a suitable modeling approach for formalizing the Financial Instability Hypothesis for two main reasons. The first reason is stressed by Bucciarelli and Silvestri (2013) and is related to Minsky's skepticism about the oversimplification required by formalized modeling and his more sympathetic view of computer simulation. Despite a robust analytical background, Minsky favored a more qualitative, rather than quantitative, presentation. According to Foley (2001), the main reason “was his recognition that the formal, statistical methods adopted by contemporary economists are inherently hostile to critical and qualitative insights into the performance of markets as human and social institutions.” On the contrary, he entertained the idea that “it might be that the most meaningful way to test propositions as to the cause and effect of financial instability will be through simulation studies, where the simulation models are redesigned to reflect alternative ways that financial instability can be induced” (Minsky, 1972). He indeed applied simulation techniques in nonlinear models inspired by the Financial Instability Hypothesis (see Delli Gatti et al., 1999, as a matter of example).

The second reason is related to the fact that AB modeling, besides involving computer simulations, are based on a bottom–up approach which appears to be particularly suitable for the formal Minskyan analysis. Indeed, the focus of the Financial Instability Hypothesis is the heterogeneity of agents' financial conditions and their evolution along the cycle. At different points of his 1975 book, Minsky stresses the need for a consistent microeconomic analysis and the consideration of agents' interaction: “an ultimate reality in a capitalist economy is the set of interrelated balance sheets among the various units” (Minsky, 20081975, p. 116).

As in traditional PK modeling, most Minskyan models are aggregative (see the surveys in Nasica, 2000; Lavoie, 2009). This modeling strategy runs the risk of missing the essential part of the story: “the composition of Ponzi, hedge, and speculative finance in any given sector can fluctuate without any changes in the aggregate balance sheet, provided that the increase in Ponzi finance is counterbalanced by improvements in the balance sheets of the remaining hedge and speculative units” (Dos Santos, 2005). As Taylor and O'Connell (1985) explicitly recognize, “shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra.”

3.1 Financial Fragility Models

The models presented here draw from the New-Keynesian literature about financial fragility which has been largely inspired by Hyman Minsky (see Bernanke and Gertler, 1989; Greenwald and Stiglitz, 1993), though without explicit recognition. For this reason, Nasica (2000) includes this stream of New-Keynesian literature in his review of Minskyan models. This literature has paved the way for the AB modeling approach to PK macro-modeling by showing the potential of AB modeling for a deeper integration of the financial sector into macroeconomic modeling, highlighting some of the complementarities discussed in Section 2, and in general by proposing AB model as a well distinguished alternative to the mainstream.

One of the most popular papers in this literature is Delli Gatti et al. (2005), which has been further developed and extended by the following Delli Gatti et al. (2007, 2010); Gallegati et al. (2007); Russo et al. (2007) among others. These models combine the core idea of Minsky's Financial Instability Hypothesis, according to which business fluctuations have a financial origin, with the New-Keynesian informational asymmetries in the credit market.

Delli Gatti et al. (2005) present an AB model version of Greenwald and Stiglitz (1993) with heterogeneous firms that interact through the credit sector. The model is substantially supply-driven with uncertainty on the demand side exogenously modeled through an idiosyncratic shock on the final goods' price. The economy is composed by a goods market and a credit market with a large number of heterogeneous firms, and one banking sector. As in Greenwald and Stiglitz (1993), firms are equity-constrained and can raise funds only on the credit market. They default when their internal finance A becomes null. Accordingly, firms set their investment expenditure, and consequently their demand for credit, as dependent on the banks' interest rates. The total credit supply, in turn, is a multiple of the banks' equity base, which is negatively affected as borrowing firms become insolvent. The total supply of credit Ls is allotted to each firm proportionally to its size and to the available cash according to the rule:

where λ > 0, Kt − 1 = ∑iKit − 1 is aggregate capital, and At − 1 = ∑iAit − 1 is the total amount of internal finance. The interest rate equates the demand and supply of credit for each firm. As a consequence, each firm will face a different cost of borrowing, which will be higher (lower) the worse (better) are the financial conditions of both the single firm and the other firms in the economy. Hence, Delli Gatti et al. (2005) devise a system of indirect interaction with a clear Minskyan flavor: as the economy grows, firms are able to fulfill their credit obligations, pushing down the interest rate and, as a consequence, raising the level of investment for all the other firms in the economy in a virtuous cycle. When the amount of debt in the economy grows, more and more firms will default, forcing the banking sector to raise interest rates causing more bankruptcies and a reduction of economic activity.

The paper provides a good example of the additional insights that an AB model can provide not only with respect to the original representative agent model by Greenwald and Stiglitz (1993), but also with respect to aggregative models. The model is stylized but nevertheless able to replicate several pieces of empirical evidence, in particular with reference to the distribution of firm-level variables. The paper also presents a discussion about how idiosyncratic multiplicative shocks and (indirect) interaction can lead to the emergence of right-skewed distributions, especially for size and growth rates, starting with uniform initial conditions.

With its simple structure, the model shows how the interaction among agents following heuristic rules can generate nonlinearities and lead to realistic aggregate results which are not directly implied by agents' behaviors. Such an outcome stresses the necessity of a holistic modeling approach which embodies fundamental uncertainty. This paper represents a good example of how AB models are well equipped to effectively incorporate these elements in their representation of the macroeconomy.

Among the subsequent developments, it is worth mentioning Delli Gatti et al. (2010), which extend this approach by allowing for heterogeneity in the banking sector and introducing capital goods producing firms, thus enabling agents to directly interact. The economy is modeled as a multilayered network in which firms have connections with credit suppliers and with input suppliers. The formation of links in this network is random: each goods producing firm can see only a subset of potential suppliers of inputs and credit and will try to obtain inputs or credit from those among them who offer the best conditions. The endogenous network formation provides additional detail with respect to Delli Gatti et al. (2005) for the study of financial contagion and its effects on the phase transitions in the business cycle.

The model is able to replicate empirical evidence about firms' distributions. As for the network, the random matching mechanism affects the degree distribution making big crises more likely when the credit market is more concentrated.

The network representation of a financial fragility model makes more evident the role of institutional structures and provides a more detailed representation of the complex chain of feedback effects that determine the aggregate outcomes. The fundamental uncertainty in which single units operate is represented by the matching mechanism: agents can see only a random subset of possible matches. As shown in the reminder of the paper, this type of matching mechanism is a common feature in AB models with networks.

3.2 Minskyan Models with Different Classes of Agents

This second group of papers within the Minskyan tradition uses AB modeling to link the business cycle to the density of agents in the different groups identified by Minsky (hedge, speculative, and Ponzi units).

One of the first contributions in this area is the AB model by de Freitas and Lima (2007). They propose a model with a fixed number of firms, which form expectations about future demand according to a predetermined set of possible rules and consequently determine their need of funding to finance production with labor as the only input. Aggregate demand follows an exogenous autoregressive process. It is assumed to be the sum of a number of demands (all of equal amount) of consumers who randomly match with firms on the goods market. Consequently, firms may accumulate unwanted inventories. Each firm has a specific markup and the firms with the highest leverage can imitate the pricing behavior of f randomly surveyed competitors.

As Foley (2001) remarks, one of the central issues in modeling Minsky's theory of business cycles is the formalization of the mechanism that leads the financial sector to expand the supply of finance during upturns and to restrict it in the subsequent downturns. In de Freitas and Lima (2007), money is fully endogenous and the banking sector applies a variable mark-up to the policy rate. In particular, banks calculate a risk-adjusted mark-up which depends on the previous period's default rate, implementing an indirect interaction mechanism similar to the one in Delli Gatti et al. (2005). Formally, the mark-up h at time t is given by

where d is the percentage of defaulted debt.

Firms are classified as: hedge if their cash flow (generated by interest accrued on deposits and sales revenue) is enough to repay both the interest and the principal of debt, speculative if the cash flow is big enough to repay at least the interest, and Ponzi if they need to rollover both the interest and the principal component of debt.

Through computer simulations, de Freitas and Lima (2007) test the dependence of systemic financial fragility (measured by the proportion of the three types of agents) on the parameters of the model, for example, on the effectiveness of the matching process in the goods market and the sensitivity of firms' price mark-up on macroeconomic conditions. Simulations show how the distribution of firms across the three financing regimes is sensitive to the flexibility of price mark-ups (proxied by the parameter f) and the size of the sample of firms surveyed by consumers. The output and credit cycles appear to be strictly correlated but the distribution of firms for financing regimes does not display any pattern.

Despite the fact that sensitivity study involves factors that are not central in Minsky's narrative (such as informational asymmetries and local imitation), and the investigation of financial fragility does not clarify its genesis and diffusion process, this paper is an interesting pioneering attempt to explore the potential of AB modeling in a Minskyan model.

As in Delli Gatti et al. (2010), a random matching mechanism and local interaction are used to represent the fact that agents face an uncertain and evolving environment and adopt heuristic rules.

Chiarella and Di Guilmi have developed the above line of research in three different papers (Chiarella and Di Guilmi, 2011, 2012b, 2017). Chiarella and Di Guilmi (2011) adapt in an AB framework the aggregative Minskyan model by Taylor and O'Connell (1985). Firms are heterogeneous in size and financial condition, and are subjected to idiosyncratic shocks.

Two contributions of this paper are worth emphasizing. First, the model is solved both numerically and analytically, following the aggregation method proposed by Aoki and Yoshikawa (2006) and Di Guilmi (2008). This method is particularly suitable for representing the Minskyan dynamics of the business cycle as it groups agents in clusters according to measurable characteristics. In order to keep the derivation as simple as possible, Chiarella and Di Guilmi (2011) reduce the number of categories to two, grouping together speculative and Ponzi in the classification of de Freitas and Lima (2007).

The second contribution is the procyclicality in the financial sector's propensity to lend through a Tobinian system which determines the total amount of wealth in an endogenous money setting.3 The prices of the two types of equities (stocks issued by hedge and by speculative firms) are quantified as clearing market prices. Capital gains (losses) determine the availability of credit and, consequently, the lending interest rate.

As shown by both simulations and analytical solution, asset price booms generate cheaper credit and more investment in a virtuous cycle, which is also the cause of growth in the proportion of speculative firms. The share of investors' portfolio allocated to hedge or speculative firms' equities depends on a stochastic mechanism: when a series of shocks reduces the proportion of wealth invested in speculative firms' shares, their financial condition worsens. The most financially distressed firms will default, reducing investors wealth and setting the stage for a further worsening in credit conditions. The higher cost of credit will squeeze firms' investment and the previous virtuous cycle is reversed.

The paper provides a microeconomic explanation of macroeconomic phenomena in line with Minsky's theory, rooted in a complexity perspective. During booms, the capital and the debt of speculative firms grow faster until the system hits a critical leverage threshold, while hedge firms have a steady rate of growth over time. Hence, the variation in the balance sheet structure of speculative units is at the root of short-term fluctuations and changes in firm size distribution over the cycle.

Chiarella and Di Guilmi (2012b, 2017) extend this framework using Minsky's original classification of firms into three categories and solving the model only numerically. In particular, Chiarella and Di Guilmi (2012b) add a government sector that runs anticyclical fiscal policy, and Chiarella and Di Guilmi (2017) also include a central bank that sets the reference interest rate with different types of Taylor rules. Consistently with PK theory,4 they show that an active fiscal policy can curb the amplitude of the boom-bust cycles whereas an active monetary policy can possibly generate undesired effects due to the complex chain of effects on firms' behavior generated by a change in the policy rate. Chiarella and Di Guilmi (2017) also find that endogenous credit can contribute to keep goods price inflation low during stock market booms. In both papers, the simulations reveal a strict correlation between the cycle and the evolution of the densities of agents in the three groups, and the consideration of the three financial regimes adds further detail to the analysis of the joint dynamics of firms' financial fragility and the macroeconomy.

4. AB Model Applications in Structuralist-Neo-Kaleckian and Evolutionary Models

As argued by Setterfield and Gouri Suresh (2016), AB models are suitably equipped to deal with path-dependency in a Keynesian and Schumpeterian sense. Their argument relies on three observations. First, AB models represent the economy as a complex system with nonlinearities and high sensitivity to initial conditions. Second, two of the defining features of AB models are agent heterogeneity and interaction, which are also found in different definitions of path-dependence (such as hysteresis and lock-in). Third, AB models are open-ended and their final outcome can only be identified through computational techniques (Arthur, 2013).

Following Setterfield and Gouri Suresh's suggestion, this section groups together AB models developed by Giovanni Dosi and his research group at Sant'Anna in Pisa, Italy, which combine a Schumpeterian evolutionary approach with some PK characteristics (presented in Subsection 4.1), and structuralist and Neo-Kaleckian models with AB features developed by Mark Setterfield and coauthors (presented in Subsection 4.2).

4.1 Evolutionary Models with PK Features

Dawid (2006) provides a review of AB models with technological change and innovation. Given its nature and scope, AB modeling has been widely used by evolutionary economists. For the purposes of this review, the series of models defined as “Keynes-Schumpeter” is of particular interest (Dosi et al., 2008, 2010, 2013, 2015). This type of models has proved to be able to replicate a large number of stylized facts and to be flexible enough to be adapted in different contexts (see, for example, Dosi et al., 2016, for an assessment of job market reforms). These models originally combine an evolutionary approach with technological change and some elements of PK macroeconomics. In particular, they merge an evolutionary approach (Nelson and Winter, 1982) with Keynesian effective demand to answer the question “How does aggregate demand modulate the diffusion and the macro-impact of technological innovations?” (Dosi et al., 2010).

Whereas some elements of this framework are not PK, for example, the supply-side determination of innovation,5 the most recent versions (such as Dosi et al., 2013) embody some essential PK elements and, interestingly, achieve PK conclusions in terms of policy. The elements that are related to the traditional PK literature are the Minskyan credit cycle and its effects on the macroeconomy (which is modeled in line with Delli Gatti et al., 2005) and a Kaleckian modeling of the functional distribution of income, which depends on the firms' price mark-up.

The latest versions of this model present two industrial sectors (capital goods and consumption goods), a banking sector with heterogeneous banks that set their lending rates depending on the risk class of their borrowers, a government that collects taxes and decides the level of expenditure and the unemployment benefits, and a central bank that sets the policy rate to steer the economy toward its inflation and unemployment targets.

The result is a fairly large and complex model. However, the results of the simulations are neat and the authors provide an in-depth sensitivity analysis of the relevant parameters. Dosi et al. (2016) enrich the previous papers by studying the effects of joint variations in the core parameters thanks to recent techniques of global sensitivity analysis for numerical systems.

The results of the simulations in Dosi et al. (2013) identify fiscal policy as the best tool to smooth the cycle and reduce inequality. The microeconomic detail enriches the evaluation of policy, showing that fiscal policy has a greater impact the larger is the degree of inequality in income distribution. In particular, an increase in the tax rate and in the size of the unemployment benefit as a percentage of the average market wage reduce the standard deviation of GDP growth rates, make full-employment more likely, and reduce the probability of a crisis. These effects are stronger when the functional distribution of income is more skewed toward wages. A well-designed fiscal policy can also foster innovation and prevent the economy from falling into long-term stagnation. Overall, the numerical results prove that austerity measures are self-defeating and that monetary policy should abandon interest rate targeting in order to complement and support the stabilizing effort of the government.

4.2 Kaleckian Models with AB Features

Different from the other models surveyed in this paper, which are constructed in a bottom–up fashion, Suresh and Setterfield (2015) (but see also Gibson and Setterfield, 2015a, 2015b) build a standard aggregative structural model and then reformulate only the relevant equations to account for heterogeneity in the firm sector. Building on Setterfield and Budd (2011), the path-dependency of the economic system is investigated by Suresh and Setterfield (2015) focusing on the “state of long-run expectations” as the reaction of agents to fundamental uncertainty in the Keynesian sense. In particular, firms heuristically revise upward (downward) their investment strategy looking at their capacity utilization in the previous unit of time according to a composite criterion whose main determinant is

numbered Display Equation

where ujt − 1 and ut − 1 are the capacity utilization for, respectively, the single firm and the whole economy, and κ, c are constant.6 The parameter 0 ⩽ κ ⩽ 1 quantifies the “degree of isolation” of a firm. Firm j uses adaptive expectations about its future capacity utilization: uejt = ujt − 1.

In the simulations, an idiosyncratic shock is exogenously introduced in the capacity utilization in the second period in order to measure the magnitude of the standard deviation of the distribution of capacity utilization as a function of the parameter κ. The numerical results suggest a nonlinear relationship which displays a reverse-U shape, with critical points located at κ = {0.4, 0.85}.

The network effects of financial constraints are investigated by Gibson and Setterfield (2015a). They focus on the short run and, in particular, the role of financial constraints for firms when lending agents have heterogeneous expectations (namely, bull or bear). The aim of the paper is to show the possible additional insights on the real-financial interaction coming from integrating AB modeling into a structuralist perspective.

Gibson and Setterfield (2015a) build a static network to connect financial agents among themselves and with firms. Firms make decisions about investment on the basis of their capacity utilization and profit rate. They consequently determine the level of aggregate demand, which is allocated to each firm with an algorithm that ensures that for each firm capacity utilization is strictly lower than 1. When the value of investment is higher than retained profits firms demand credit from the lender (financial agent) to which they are connected. The lender uses a Bayesian procedure to update her forecast. Looking at the forecast and the availability of capital (directly or through lending from other financial agents), the lender decides whether to approve the loan requested by the firm. In case of a negative decision, the firm's investment is not realized. Financial networks are either random or subject to preferential attachment, and either weighted or unweighted by shares of capital stock.

In contrast to other models of real-financial networks (as the cited Delli Gatti et al., 2010, among others), in this paper a more interconnected system appears to be more resilient to financial shocks and less prone to crashes. This result is probably affected by the mechanisms of link formation and the consequent transmission of financial distress. In both weighted and preferentially attached networks, the number of loans denied is larger. However, the mechanism of link formation does not affect GDP growth, only the distribution of growth rates: large firms grow faster due to their easier access to credit. Financial crashes are more likely to occur in a network with preferential attachment.

This set of models represents a hybrid between purely structural aggregative models and AB models, built through a sort of disaggregation process. Different from typical AB models, in the simulations some results are subject to further ex post analytical manipulations before being fed back to the model and as a consequence it is not clear whether the final macroeconomic results can genuinely be considered emergent properties.

5. SFC-AB Models: Toward a New Benchmark?

In recent years, stock-flow consistency has become more and more popular in macroeconomic AB models with PK features.7 At the same time, PK scholars have started microfounding their SFC models using an AB approach.

5.1 SFC-AB Models with PK Features

Seppecher (2010) and the following Seppecher and Salle (2015) present a SFC-AB model, called Jamel (Java Agent-based Macro-Economic Laboratory), in which the phase transitions during business cycles are determined by the opinion dynamics, as modeled in De Grauwe (2008). In particular, consumers and firms can be optimistic or pessimistic about, respectively, their consumption decisions and the targeted level of leverage. The model combines non-PK elements (for example, a Calvo pricing mechanism for price adjustment) together with PK tenets. Credit supply is perfectly elastic at a fixed interest rate, which is increased to a constant higher rate when a borrowing firm is forced to rollover its debt.

Each agent (household or firm) decides about her attitude looking either at her own past situation with a predefined probability 1 − p or to a given subset of neighboring agents with a probability p. The simulations in Seppecher and Salle (2015) produce realistic dynamics of a business cycle, which is driven by endogenous waves of optimism and pessimism among agents: market sentiment pushes up consumer spending and distributed profits, leading to a boost in aggregate demand that increases the level of employment and profits. However some agents remain pessimistic and can spread their pessimism to other agents, depending on the size of the parameter p. If this contagion reaches a critical threshold, the virtuous cycle is reversed and the economy enters into a recession, with reduced consumption and deleveraging.

The paper enriches the analysis of financial fragility with original insights about the role of opinion dynamics in the propagation of shocks through random networks. Network effects can be clearly assessed through the sensitivity study of the parameter p, which reveals that for stronger imitation effects the volatility of the output gap generally increases because of the faster transmission of changing sentiment among consumers.

Bruun provides a few early attempts of SFC-AB models that are generically qualified as Keynesian. Among these, Bruun (2010) is more directly related to traditional SFC models, and proposes a representation of a three-sector economy with producers of investment goods, producers of consumption goods, and consumers. One of the most original features of the model is that the distributional effects are investigated using thresholds for negative wealth (below which households reduce consumption) and positive wealth (above which households increase consumption). In this setting, the sensitivity study reveals that the exogenous creation of inequality (low threshold for poor household and high threshold for rich household) generates faster growth but also sets a limit for it due to the transfer of wealth to the upper tail of the distribution. Unfortunately, these results are not further investigated and related with the other similar contemporary studies on this topic.

Riccetti et al. (2015) introduce stock-flow consistency in a model that borrows from Riccetti et al. (2013) and from Delli Gatti et al. (2005, 2010). Although the model is, strictly speaking, not PK (for example, money is not endogenous), it is worth mentioning here because in this model business fluctuations are due both to Minskyan financial dynamics of the type seen in Delli Gatti et al. (2005, 2010) and to original Goodwin-type effects originating in the labor market. The economy is composed of a set of heterogeneous firms, households and banks, together with a central bank and a government. Agents randomly match in the goods, credit and labor markets and heuristically revise their behavior accordingly to their situation in the previous unit of time. In particular at time t, a worker revises up (down) her satisficing wage if she was employed (unemployed) at time t − 1. This simple behavioral rule creates Goodwin dynamics that interact with the credit market conditions. Banks linked to defaulted firms lose a fraction of their loans while defaulted firms are replaced by new ones with a fresh endowment of net worth and a targeted leverage ratio of 1.

The realization of Goodwin cycles as an emergent property based on a heuristic behavior at the agent-level adds an interesting perspective for the study of this type of dynamics. The bargaining power of the workers is differentiated across individuals and becomes endogenous. This allows for the identification of more specific policy prescriptions, such as the government's acyclical hiring of workers. From the simulations, the authors conclude that this type of fiscal policy is effective at stabilizing the economy. The simulations also highlight the correlation between debt cycles and business cycles, and in particular the amplifying effect of financial factors on the business cycle.

Russo et al. (2016) further develop this framework by introducing a consumption function with habit formation. This extension allows the authors to shed light on the empirical finding by Cynamon and Fazzari (2013) and Perugini et al. (2016), among others, that the redistribution of income toward the top earners observed since the 1980s has not resulted in stagnation due to the increase in the availability of consumer credit and the consequent use of leverage to finance consumption by the poorest households. Using two sets of simulations (with and without consumer credit), the model reproduces one of the main empirical findings of the cited papers: the availability of credit and the resulting increase in leverage make financial crises more likely. As a consequence, policy makers face a trade-off between stability and fast growth. Higher leverage is at the root of wealth inequality, because of the growing interest burden faced by borrowing households, and interestingly, Monte Carlo replications show that the presence of consumer credit has a negative effect on average output employment in each run.

The Eurace model (Raberto et al., 2012) is a very articulated and large AB model which aims to replicate in detail the features of a real economy. The version presented in Raberto et al. (2012) consists of a goods production sector, a capital producing firm, a household sector, a banking sector, a government, and a central bank. The model presents some essential PK features, a demand-driven goods market, a standard mark-up pricing rule, and endogenous money,8 together with some significant departures from the typical PK model architecture, such as a Cobb–Douglas production function for firms in the consumption goods sector and the determination of investment on the basis of a cost-minimizing approach. Another difference with a standard PK model is the consumption decision, modeled as buffer-stock saving behavior (Carroll, 2001).

Firms' expectations about future demand are quantified using a weighted average over the past periods, considering the amount of inventories. The matching in the goods, labor, and credit markets occurs after a local search over a random subset of counterparts. Once consumption for household h is determined, the allocation among the different firms f obeys the following logit model:

where Λ is a parameter and pf is the price of output for firm f (given that firms have different technology, they will also have different costs and prices).

Firms face two types of bankruptcies: insolvency bankruptcy and illiquidity bankruptcy. In the former, equity goes to zero and debt is restructured, forcing the defaulted firm to adopt a lower leverage target. In the latter, firms still have some residual equity. In both cases, firms do not disappear but simply cease production and try to raise new capital in the financial market in order to restart their activity, ensuring the accounting consistency for stocks and flows. The central bank follows an inflation targeting rule.

The simulations reveal a Minskyan pattern in which the debt cycle is at the root of the business cycle. In particular, the regulatory rule for banks causes a credit squeeze at the peak of the cycle when the more leveraged firms default, causing a reduction in capital to the lending banks. Computational experiments with different regulatory constraints confirm the dependence of the real economy on the credit market.

As argued by Lengnick (2013), the degree of complexity of very large AB models, while allowing a comprehensive representation of an economic system, makes it problematic to identify the causal relationships and thus to unambiguously interpret the results, justifying the concerns expressed by Foley (2017) for AB modeling. However, Raberto et al. (2012) can provide an example of how an SFC representation can to some extent discipline the construction of such a large model by limiting the number of modeling options.

This framework has been subsequently used by Erlingsson et al. (2014) to study the real effects of a housing market bubble. Dawid (2015) and Dawid et al. (2016) propose a policy-oriented variant of this model, aimed at a specific application to the European Union economies.

5.2 PK SFC-AB Models

Kinsella et al. (2011) is probably the first paper with a typical SFC model in an AB setting. The model consists of four sectors: firms and households, which are modeled at agent-level, banks, which are treated as an aggregate sector, and the government.

Households have a limited life-span (75 years) and can accumulate wealth, through savings, and increase their job skills, through debt-financed education and learning-by-doing. After 75 periods, each household is replaced by a newborn one, which inherits its parent's wealth but not the working skills. Firms spend their profit in either expanding their productive capacity or, alternatively, engaging in risky innovation according to a fixed probability. The matching on the labor market occurs randomly. The offered wage depends on the financial resources of the employer and on the level of education of the potential employee; the demanded wage depends on the household's wealth, skills, and education plus a random component quantifying the relative bargaining power. Money is endogenous as the banking sector elastically supplies credit to households and firms (even though at a variable rate that depends on the loans' profitability). However, the accumulation of debt is limited as agents are bankrupted when their negative money reaches a given threshold.

To ensure stock-flow consistency, defaulted firms and households create bad debt for banks. Bankrupted firms are immediately replaced and their initial endowment is supplied by the government at no cost. Aggregate demand is allocated to each firm according to an index based on the productivity of its workers and its productive capacity.

Simulations of the model show fairly realistic dynamics of the macroeconomic variables and a Pareto distribution of income for the top 5% with a Pareto parameter close to the one empirically detected. The bottom 95% are distributed according to a gamma or exponential distribution.

Finally, it should be noted that the conclusions of Kinsella et al. (2011), stressing the negative macroeconomic consequences of unfettered market in terms of “winners” and “losers,” are close to the political core of PK economics (Lavoie, 2014) discussed in Section 2.3.

The authors show that the AB modeling approach allows for a more realistic and complex representation of an evolving economy with inter-generational dynamics than the one possible in aggregative (or in representative agent) models. The simulations generate a realistic distribution of household income, and the authors provide an explanation for this piece of evidence based on statistical physics (Foley, 1994). The authors do not further explore this avenue to study, for example, the dynamics of the distribution over the cycle or possible policy implications.

The latest generation of this class of models is explicitly identified as SFC-AB models in order to stress the continuity with the two original literature. They are often the result of collaborations between scholars from both camps (Godin, 2015).

Carvalho and Di Guilmi (2014) and Di Guilmi and Carvalho (2017) develop microfounded neo-Kaleckian models to study the joint dynamics of leverage, inequality, and the business cycle. The originality of these contributions is their use of the standard SFC structure, conventions, and analysis, but built in a bottom–up fashion, starting with the microeconomic behavioral equations.

Both papers present SFC-AB models that are analytically solved by the master equation techniques introduced by Aoki and Yoshikawa (2006) and Di Guilmi (2008) in order to derive the aggregate equations of the SFC model from the agents' behavioral rules. The result is a dynamical system that includes micro-level variables; the analysis of steady state and stability is used to integrate and interpret the numerical outcomes of the Monte Carlo simulations. Consequently, the social accounting matrix and the SFC dynamical system feature micro-level variables. Both models are closed in a Keynesian fashion by the functional identification of a multiplier that allows for the contemporaneous determination of aggregate demand and demand for labor.

Carvalho and Di Guilmi (2014) study the interaction of wealth and income inequality, functional distribution of income (which is exogenous and constant) and the business cycle proposing an SFC-AB model on the household side while the firm side is modeled as an aggregate as in the traditional PK-SFC literature. The sensitivity study of both the numerical results and the analytical solution highlights the effect of the functional distribution of income on the cycle. In particular, a higher share of profits on total output makes the system more volatile and the personal distribution of income more skewed.

Di Guilmi and Carvalho (2017) complement their previous paper by modeling the production side as an AB model while keeping the other sectors of the economy as aggregates. For analytical tractability, they use a stylized version of Minsky's classification criteria dividing firms in hedge and speculative and not allowing for Ponzi behavior. Aggregate demand is allocated according to firms' size with a stochastic preferential attachment shock that partially redistributes demand across firms. Firms with a leverage ratio higher than a fixed threshold are bankrupted and their capital is reallocated at zero cost to continuing and newborn firms.

Both numerical and analytical results investigate the effect of firms' behavioral parameters on the stability of the system, in particular a more conservative attitude of hedge firms has the undesired effect of making the expansionary phase more reliant on speculative firms and, therefore, more unstable. A functional distribution of income more geared toward profits has a destabilizing effect on the economy.

By integrating statistical mechanic and AB modeling approaches, these papers indicate a possible avenue for overcoming the identification issue stressed by Foley (2017). Thanks to the analytical solution, the steady state of the model can be determined and the conditions for its existence identified. Moreover, since the macroeconomic variables can be expressed as a function of microeconomic quantities, the causal links can be clearly analyzed. Given its analytical complexity, applications of this method may face a trade-off between tractability and degree of sophistication.

Michell (2014) presents an SFC-AB model to investigate the Steindlian idea of the natural emergence of oligopoly in competitive markets, and the consequent fall in the rate of investment and modifications in the functional distribution of income, which is therefore endogenous in this model. The paper proposes a Kaleckian growth model in which firms set their targets for output, fixed capital investment, and loans. As in Di Guilmi and Carvalho (2017), the firm sector is microfounded while the household and banking sectors are modeled as aggregates. Due to the numerical solution used by Michell (2014), the model presents a different closure from Di Guilmi and Carvalho (2017): the macroeconomic variables are either obtained as a sum of the microeconomic quantities in the productive sector, or quantified by means of a system of equations in a typical SFC fashion. As such, output and aggregate expenditure (households' consumption plus investment) may differ. Aggregate demand is allocated to each firm using a mechanism similar to Di Guilmi and Carvalho (2017), with an additional parameter to govern the proportions of demand allocated according to size and stochastic shock, respectively. Michell (2014) shows that, when the size of the shock is relatively small, the model confirms Steindl's intuition: starting from a uniformly distributed capital endowment, market concentration rapidly increases, the functional distribution shifts toward profits, and the rate of investment falls. However, the size of the shock is probably too big for a fair assessment: when the stochastic element accounts for 30% of the criterion for demand allocation, the dispersion of firm-level variables is unrealistically large.

The author demonstrates the potential of AB microfoundations for SFC analysis from two different perspectives. First, it highlights the micreconomic behavioral and structural details of the joint evolution of market concentration, leverage, and investment that cannot be captured when intrasectoral flows are not modeled (as in standard aggregative SFC models). Second, the AB microfoundations enable the model to reconcile the procyclicality of debt implied by Minsky's Financial Instability Hypothesis with the Kaleckian theory of procyclicality of profits, showing that while the proportion of firms in financial distress is rising, the net financial position of the productive sector improves thanks to the accumulation of cash by the largest units.9

Caiani et al. (2016) propose an SFC-AB model that aims to be general enough to define a benchmark for this class of models. The model is composed of a large number of heterogeneous households, firms (divided in capital goods and consumption goods producing firms), banks, a government sector, and a central bank. The agents interact in different markets (consumption goods, capital goods, labor, credit, and deposit) through a random matching mechanism borrowed from Riccetti et al. (2015). The duration of bank loans is set at 20 periods, and in each period the borrowers pay the one-period interest plus a share of the principal. Since a firm resorts to borrowing finance (randomly matching with a bank) whenever the operating cash flow is not enough to face its expected financial commitments, at a given point in time each firm can have multiple active loan contracts.

Firms and banks can default when their equity is wiped out but, in order to maintain their number constant, they are bailed in by the households who own their capital, and by depositors. Money is endogenous but, as in Riccetti et al. (2015) and Dosi et al. (2013), credit rationing may occur as a consequence of a prudent behavior of banks. In particular, the supply of credit depends on the capital ratio of the lending bank and the riskiness of the borrower, measured by the probability of default during the life of the loan and its operating cash flow and value of collateral.

The main contribution of Caiani et al. (2016) is methodological as they identify the crucial issues for the development of consistent and usable SFC-AB models and propose original solutions to them. The most important contribution concerns the calibration and accounting validation of the model. As Caiani et al. (2016) correctly remark, many of the existing SFC-AB models have a consistent structure of stock and flows as defined by their behavioral and accounting equation but, when it comes to the implementation of the numerical simulations, the initial calibration may not respect this accounting consistency. Their solution consists in: first, deriving an aggregate version of the model; second, identifying a steady state in which real variables are constant and nominal variables grow at a constant rate; third, numerically solving the model for the initial conditions and some of the parameters by considering as exogenous parameters those for which empirical estimates are available and those that the authors want to control. Furthermore, the initial duration of loans10 and the matching in the various market is set in order to ensure the accounting consistency.

This innovative calibration procedure allows the authors to analyze the time series produced by the simulations from the beginning and not after a burn-in phase, as is standard practice in AB models. Caiani et al. (2016) argue against this practice because of its arbitrariness and the strong path-dependency of AB models.

The numerical simulations of this model are run by means of a platform appositely developed for AB model-SFC, named Java Macro Agent Based (JMAB) tool suite. The results of the model are in line with a number of stylized facts as detected by the empirical literature. The time series of macro-variables match their empirical counterparts for volatility, auto- and cross-correlation as do the distributions of firm-level variables (size, growth rates, bad debt, and bankruptcies). As for the network structure, the degree distribution for banks is right-skewed, and the number of degrees per bank is higher than the number of degrees for firms. The analysis of the results is completed by a robustness check, which demonstrates that these outcomes are not determined by the choice of the parameter set.

6. Concluding Remarks

This paper provides an overview of the literature about PK-AB modeling and AB models with PK features. Within this literature, four different streams can be isolated: Minskyan AB models; evolutionary AB models with PK features; AB models belonging to the neo-Kaleckian and structuralist traditions; and SFC-AB models. In surveying the various papers, it is argued that this cross-fertilization can benefit and enrich both AB and PK modeling approaches.

The construction of a microfounded PK macro-model faces three main challenges. The first is the elaboration of microbehavioral rules consistent with the traditionally aggregative structure of PK macro-models. As mentioned in the introduction, aggregative PK models often use microeconomic behavioral rules to infer macroeconomic relationships. Consequently, most of the models surveyed here manage to adapt the PK aggregate equations at the agent level, introducing in some cases specific micro-variables. In some instances, when the microeconomic detail was not essential to the analysis, sectors have been kept as aggregate.

The second issue is twofold and concerns, on the one hand, the computation of the relevant aggregate quantities and, on the other hand, the allocation and matching in markets where only aggregate quantities are known. The former is usually solved by devising a time-structure that allows for the determination of the variables in a logically consistent manner. Only in one instance the usual Keynesian closure with the multiplier is adopted. For the latter issue, a range of possible criteria (from size to price to uniform distribution) have been proposed.

Finally, SFC-AB modeling faces two main specific types of issues. The first consists in modeling agents' demography: what happens to the assets of bankrupted agents and where does the endowment of newborn agents come from? In most cases, the issue is avoided by keeping the number of agents constant with different mechanisms: immediately replacing defaulted firms, keeping them temporarily inactive or bailing them in. Only in one instance the replacement is not one-to-one and stock-flow consistency is ensured by the redistribution of firms' assets to continuing firms and new entrants. The second type of issues specific for SFC-AB modeling is how to ensure the consistency of stocks at any stage of computer simulations. While all the surveyed models present a structure of behavioral equations that ensures the consistency of stocks and flows (verified for the models in Subsection 5.2 by a social accounting matrix), Caiani et al. (2016) is so far the only paper in which this problem is explicitly discussed, providing an original initialization algorithm that ensures the accounting consistency from period zero of the simulations.

The microfoundation through an AB approach provides a number of potential insights. The review shows that it is possible to associate the evolutionary dynamics of agents to the different phases of the business cycle and to identify which microeconomic conditions (agents behaviors, endowments, or financial situations) guarantee faster or sustainable growth. Policy analysis is enriched in two main ways. First, a policy measure can be differentiated by classes of agents (for example, unemployed workers) and its impact can be assessed not only with reference to the macroeconomy but also to single social classes, groups, or individuals. Moreover, the distributional effects of a policy can be evaluated in full detail. Second, in assessing a particular policy, AB modeling can shed light on the unintended or undesired consequences that can be generated by the complex interaction of agents. As for the social structure, AB models with networks are able to investigate the aggregate outcomes generated by different types of social norms and institutions (represented, for example, by different types of connectivity, information flows, and mechanisms of link creation).

AB modeling can benefit from the cross-fertilization because PK economics can represent a benchmark for limiting the degrees of freedom in building the model and a reference for the interpretation of the results addressing the concerns expressed by Foley (2017). These concerns appear to be legitimate in particular for large-scale models. A possible solution is a deeper integration between AB models and statistical mechanics, which can also open new directions of research for PK economics.

Ackowledgments

The author wishes to thank Timo Henckel, Alberto Russo, and two anonymous referees for helpful comments. The usual caveats apply.

Notes

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