by Paul Beaudry, Dana Galizia and Franck Portier
Flocking birds is a famous natural life example of emergent phenomena —i.e. aggregate phenomena whose properties are very different from the ones of the individuals involved, when taken separately. There has been little effort in economics at understanding emergent phenomena in a systematic way. The existence of a two-way feedback between microstructure and macrostructure has been recognised for a very long time, from the invisible hand of Adam Smith  to the work of Hayek  and Schelling . There are many examples where aggregate outcomes may display instability when individual level decisions are aimed at favouring stability, but it is unclear what are the key forces driving such phenomena.
In our work, we aim at filling this gap in two ways. First, we study the properties of a reduced form dynamic model that can be seen alternatively as the simplest Agent-Based Model or as the simplest Dynamic Stochastic General Equilibrium model in which the assumption of rational expectations can be crucial. Second, we dig into the micro-foundations of emergent phenomena in macro by studying « dynamic Cooper & John’s economies », which are economies with strategic interactions and accumulation. In this blog post, we successively describe the approach and results we obtained in these two projects.
How did modern macroeconomics addressed the issue of macroeconomic fluctuations?
Dynamic Stochastic General Equilibrium models (See Kydland & Prescott  for pioneering work and Smets & Wouters  for more recent New-Keynesian incarnation) typically see the macroeconomy as being smoothly converging to a stable steady state absent of shocks, and fluctuations are created by exogenous shocks to fundamentals. In that sense, there is no qualitative jump between individuals and the aggregate in the most standards incarnations of those models.
Although the steady state is stable absent of shocks, there is a possibility of belief shocks (sunspots) to affect the economy when agents have rational expectations and the equilibrium is indeterminate, as introduced by Cass & Shell , Azariadis  and Benhabib & Farmer . The assumption of rational expectations is then crucial to generate such emergent self-fulfilling fluctuations.
Agent Based Models (See Schelling  for a very first application to social sciences and Delli Gatti et al  for an account of Agent Based modelling in economics) is another route taken by macroeconomists to bridge microeconomic behaviours and macroeconomic outcomes. In these models, autonomous agents are interacting with bounded rationality and not fully rational expectations, and the aggregate properties of the model cannot be directly inferred from micro behaviour, leading to the emergent phenomena such as boom-bust cycles and hysteresis.
In this research, we study what we believe is the minimal model that one can write to generate emergent aggregate outcomes. It features accumulation, interactions between agents and possibly (rational) expectations. In the model, agent i takes actions and accumulates according to
and we study symmetric equilibria when a large number of agents are interacting. Key elements in the model are accumulation X_it, sluggishness in actions e_(it-1) expectations E_it[e_(it+1)] and crucially the term E_it[F(e_t)] that embeds social interactions between agents, with substitutabilities when F'< 0 and complementarities when F'> 0.
The configuration α_3 = 0 can be interpreted as a simple dynamic agent-based model, while when α_3 ≠ 0, one could see the model as a reduced form of a DSGE (See Beaudry, Galizia & Portier [2015,2018] for a formal derivation in two different settings).
Our results are that complementarities typically create emergent phenomena, while substitutabilities do not. Then, depending on the sign and size of α_1, α_2 and α_3, hysteresis, limit cycles or sunspot equilibria can emerge. Each of these configurations can be related to economic properties of the model.
In their seminal work, Cooper & John  showed how coordination failures can be a foundation for « Keynesian » macroeconomics. In their static framework, strategic complementarities could indeed lead to inefficient equilibrium, multiple equilibria and policy multipliers. Since that paper, emphasis has been put on the the dynamic properties of macro models, and numerous business cycle models has been written in which coordination failures create emergent equilibrium (indeterminacy in Benhabib & Farmer , hysteresis in Taschereau-Dumouchel & Schaal , limit cycles in Beaudry, Galizia & Portier ).
In this paper, we want to adopt an abstract setup à la Cooper & John , put it in a dynamic framework and study the emergent properties of equilibrium as a function of the payoff structure (the « V function »). To do so, we study a model in which agent maximises
subject to the accumulation equation
and the transversality condition
with X_i0 given. Again, we study symmetric equilibria when a large number of agents are interacting.
Our results can be best described as follows. First, complementarities between actions favour determinate hysteresis and cycles, substitutabilities between actions favour determinacy and stability. Second, interactions between aggregate action and individual stock (complementarities or substitutabilities) favour indeterminacy and hence sunspot equilibria. Third, in all the cases, whether current stock marginally increases or decreases incentives to accumulate is the key parameter to determine which type of bifurcation will occur or whether the dynamics will be oscillating or not.