A Statistical Field Approach to Capital Accumulation
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper presents a model of capital accumulation for a large number of heterogenous producer-consumers in an exchange space in which interactions depend on agents’ positions. Each agent is described by his production, consumption, stock of capital, as well as the position he occupies in this abstract space.
Each agent produces one differentiated good whose price is fixed by market clearing conditions. Production functions are Cobb-Douglas, and capital stocks follow the standard capital accumulation dynamic equation. Agents consume all goods but have a preference for goods produced by their closest neighbors.
Agents in the exchange space are subject both to attractive and repulsive forces. Exchanges drive agents closer, but beyond a certain level of proximity, agents will tend to crowd out more distant agents.
The present model uses a formalism based on statistical field theory developed earlier by the authors. This approach allows the analytical treatment of economic models with an arbitrary number of agents, while preserving the system’s interactions and complexity at the individual level.
Our results show that the dynamics of capital accumulation and agents’ position in the exchange space are correlated. Interactions in the exchange space induce several phases of the system.
A first phase appears when attractive forces are limited. In this phase, an initial central position in the exchange space favors capital accumulation in average and leads to a higher level of capital, while agents far from the center will experience a slower accumulation process. A high level of initial capital drives agents towards a central position, i.e. improve the terms of their exchanges: they experience a higher demand and higher prices for their product. As usual, high capital productivity favors capital accumulation, while higher rates of capital depreciation reduce capital stock.
In a second phase, attractive forces are predominant. The previous results remain, but an additional threshold effect appears. Even though no restriction was imposed initially on the system, two types of agents emerge, depending on their initial stock of capital. One type of agents will remain above the capital threshold and occupy and benefit from a central position. The other type will remain below the threshold, will not be able to break it and will remain at the periphery of the exchange space. In this phase, capital distribution is less homogenous than in the first phase.
Key words: Path Integrals, Statistical Field Theory, Phase Transition, Capital Accumulation, Exchange Space, Multi-Agent Model, Interaction Agents.
JEL Classification: C02, C60, E00, E1.
Heterogeneity in social values and capital accumulation in a changing world
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: In a society characterized by a multitude of heterogeneous agents and a large number of possibly immaterial goods, each one having distinct social and personal values, we study the impact of these relative values on intergenerational capital accumulation, as a function of economic and social parameters such as capital mobility, productivity and personal and social values discrepancies. Each agent is modelled by a one-period production function and a two-period intertemporal utility. Agents live, produce and consume over one period, but optimize over two periods, so providing a remaining stock of goods for the next generation. This creates a dynamics in capital accumulation depending on social and individual values. A threshold appears in capital stock accumulation that depends on personal and social values’ volatilities, and below which the initial stock will be depleted. Whereas volatility in social values increases the threshold, impairing capital accumulation, adverse shocks in goods’ values may reverse the dynamics of the accumulation process. Finally, capital mobility specifically favors forerunners, but capital accumulation in one or several sectors may shift social values in their direction, at the expense of other sectors.
Keywords: Capital theory, Capital accumulation, Investment allocation, Two-sector models, Disaggregated capital, Take-off, Threshold effect, Intergenerational models, Cambridge capital controversy.
JEL Classification: E22, O10, O30, O40.
A Path Integral Approach to Interacting Economic systems with Multiple Heterogenous Agents
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This paper presents an analytical treatment of economic systems with an arbitrary number of agents that keeps track of the systems’ interactions and complexity. The formalism does not seek to aggregate agents: it rather replaces the standard optimization approach by a probabilistic description of the agents behavior. This is done in two distinct steps.
A fi
rst step considers an interaction system involving an arbitrary number of agents, where each agents utility function is subject to unpredictable shocks. In such a setting, individual optimization problems need not be resolved. Each agent is described by a time-dependent probability distribution centered around its utility optimum.
The whole system of agents is thus de
fined by a composite probability depending on time, agents interactions, relations of strategic dominations, agents’ information sets and expectations. This setting allows for heterogeneous agents with different utility functions, strategic domination relations, heterogeneity of information, etc.
This dynamic system is described by a path integral formalism in an abstract space – the space of the agents’ actions – and is very similar to a statistical physics or quantum mechanics system. We show that this description, applied to the space of agents’ actions, reduces to the usual optimization results in simple cases. Compared to the standard optimization, such a description markedly eases the treatment of a system with a small number of agents. It becomes however useless for a large number of agents.
In a second step therefore, we show that, for a large number of agents, the previous description is equivalent to a more compact description in terms of
field theory. This yields an analytical, although approximate, treatment of the system. This
field theory does not model an aggregation of microeconomic systems in the usual sense, but rather describes an environment of a large number of interacting agents.
From this description, various phases or equilibria may be retrieved, as well as the individual agents behaviors, along with their interaction with the environment. This environment does not necessarily have a unique or stable equilibrium and allows to reconstruct aggregate quantities without reducing the system to mere relations between aggregates.
For illustrative purposes, this paper studies several economic models with a large number of agents, some presenting various phases. These are models of consumer/producer agents facing binding constraints, business cycle models, and psycho-economic models of interacting and possibly strategic agents.
Keywords: path integrals, statistical field theory, phase transition, non trivial vacuum, effective action, Green function, correlation functions, business cycle, budget constraint, aggregation, forward-looking behavior, heterogeneous agents, multi-agent model, strategical advantage, interacting agents, psycho-economic models, integrated structures, emergence.
JEL Classification: C02,C60, E00, E1
Heterogeneity in social values and capital accumulation in a changing world
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: In a society characterized by a multitude of heterogeneous agents and a large number of possibly immaterial goods, each one having distinct social and personal values, we study the impact of these relative values on intergenerational capital accumulation, as a function of economic and social parameters such as capital mobility, productivity and personal and social values discrepancies. Each agent is modelled by a one-period production function and a two-period intertemporal utility. Agents live, produce and consume over one period, but optimize over two periods, so providing a remaining stock of goods for the next generation. This creates a dynamics in capital accumulation depending on social and individual values. A threshold appears in capital stock accumulation that depends on personal and social values’ volatilities, and below which the initial stock will be depleted. Whereas volatility in social values increases the threshold, impairing capital accumulation, adverse shocks in goods’ values may reverse the dynamics of the accumulation process. Finally, capital mobility specifically favors forerunners, but capital accumulation in one or several sectors may shift social values in their direction, at the expense of other sectors.
Keywords: Capital accumulation, disaggregated capital, take-off, threshold effect, intergenerational models, Cambridge capital controversy.
JEL Classification: E22, O10, O30, O40.
From Rationality to Irrationality : Dynamic Interacting Structures
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: This article develops a general method to solve dynamic models of interactions between multiple strategic agents that extends the static model studied previously by the authors. It describes a general model of several interacting agents, their domination relations as well as a graph encoding their information pattern. It provides a general resolution algorithm and discusses the dynamics around the equilibrium. Our model explains apparent irrational or biased individual behaviors as the result of the actions of several goal-specific rational agents. Our main example is a three-agent model describing “the conscious”, “the unconscious”, and “the body”. We show that, when the unconscious strategically dominates, the equilibrium is unconscious-optimal, but body and conscious-suboptimal. In particular, the unconscious may drive the conscious towards its goals by blurring physical needs. Our results allow for a precise account of agents’ time rate preference. Myopic behavior among agents leads to oscillatory dynamics : each agent, reacting sequentially, adjusts its action to undo other agents’ previous actions. This describes cyclical and apparently inconsistent or irrational behaviors in the dual agent. This cyclicality is present when agents are forward-looking, but can be dampened depending on the conscious sensitivity to other agents’ actions.
Keywords: dual agent; conscious and unconscious, rationality; multi-rationality; emotions; choices
and preferences; multi-agent model; consistency; game theory; strategical advantage.
JEL Classification: B41,D01, D81, D82.
On Apparent Irrational Behaviors: Interacting Structures and the Mind
Pierre Gosselin, Aïleen Lotz, Marc Wambst
Abstract: We develop a general method to solve models of interactions between multiple and possibly strategic agents. Our model explains apparently irrational or biased behaviors in a person. We argue that these actions could result from several rational structures having different goals. Our main example is a model of three agents, “conscious”, “unconscious”, and “body”. Our main result states that, for an agent whose unconscious and conscious goals differ, the unconscious may influence the conscious, either directly or indirectly, via a third agent, the body. This three-agent model describes behaviors such as craving, excessive smoking, or sleepiness, to delay or dismiss a task. One of the main result shows that the unconscious’ strategic action crucially depends on whether the conscious’ actions are complementary in time. When complementary, and if the conscious is not sensitive to unconscious’ messages, the unconscious may drive the conscious towards its goals by blurring physical needs. When not complementary, the unconscious may more easily reach his goal by influencing the conscious, be it directly or indirectly.
Keywords: dual agent, conscious and unconscious, rationality, multi-rationality, consistency, choices and preferences, multi-agent model
JEL Classification: C02, C65, C70, D01, D87
A Dynamic Model of Interactions between Conscious and Unconscious
Pierre Gosselin, Aïleen Lotz
Abstract: This paper advocates that some limits of the rational agent hypothesis result from the improper assumption that one individual should be modeled as a single rational agent. We model an individual composed of two autonomous and interacting structures, conscious and unconscious. Each agent utility form depends both on external signals and other structures’ actions. The perception of the signal depends on its recipient and its grid of interpretation. We study both the static and dynamic version of this interaction mechanism. We show that the dynamics may display instability, depending on the structures interactions’ strength. However, if unconscious has a strategic advantage, greater stability is reached. By manipulating other structures’ goals, the strategic agent can lead the whole system to an equilibrium closer to its own optimum. This result shows that some switch in the conscious’ objective can appear. Behaviors that can’t be explained with a single utility can thus be rational if we add a rational unconscious agent. Our results justify our hypothesis of a rational interacting unconscious. It supports the widening of the notion of rationality to multi-rationnality in interaction.
Keywords: dual agent, conscious and unconscious, rationality, multi-rationality, emotions, choices and preferences, multi-agent model, consistency
JEL Classification: B41,D01, D81, D82
An Economic Approach to the Self: The Dual Agent
Aïleen Lotz
Abstract: This paper extends the notion of the rational agent in economics by acknowledging the role of the unconscious in the agent’s decision-making process. It argues that the unconscious can be modelled by a rational agent with his own objective function and set of information. The combination of both the conscious and unconscious agents is called the “dual agent”.
This dual agent presents rationally biased behaviors that may not disappear through aggregation, and could be potentially measured. It also provides a theoretical approach to the emotionally-driven actions. On the social sciences side, the paper advocates a wider use of hidden rationality in the understanding of human behavior.
Keywords: rational agent, decision-making, conscious and unconscious asymmetry of information, imperfect information, dual agent, theory of emotions, substantive and procedural rationality, psychology, bias
JEL Classification: B41, D01, D81, D82
The Expected Interest Rate Path: Alignment of Expectations vs. Creative Opacity
Pierre Gosselin, Aïleen Lotz, Charles Wyplosz
Abstract: We examine the effects of the release by a central bank of its expected future interest rate in a simple two-period model with heterogeneous information between the central bank and the private sector. The model is designed to rule out common-knowledge and time-inconsistency effects. Transparency—when the central bank publishes its interest rate path—fully aligns central bank and private-sector expectations about the future inflation rate. The private sector fully trusts the central bank to eliminate future inflation and sets the long-term interest rate accordingly, leaving only the unavoidable central bank forecast error as a source of inflation volatility. Under opacity—when the central bank does not publish its interest rate forecast—current-period inflation differs from its target not just because of the unavoidable central bank expectation error but also because central bank and privatesector expectations about future inflation and interest rates are no longer aligned. Opacity may be creative and raise welfare if the private sector’s interpretation of the current interest rate leads it to form a view of expected inflation and to set the long-term rate in a way that systematically offsets the effect of the central bank forecast error on inflation volatility. Conditions that favor the case for transparency are a high degree of precision of central bank information relative to private-sector information, a high precision of early information, and a high elasticity of current to expected inflation.
Interest Rate Signals and Central Bank Transparency
Pierre Gosselin, Aïleen Lotz, Charles Wyplosz
Abstract: The present paper extends the literature on central bank transparency that relies on information heterogeneity among private agents in four directions. First, it adds the interest rate to the list of signals that the central bank can reveal. Second, it allows for more than one economic fundamental. Third, it extends the range of uncertainties that matter. So far the literature has focused on uncertainty about the economic fundamentals, assumed to be estimated with known precision; we also allow for uncertainty about precision. Fourth, it derives results that are general in the sense that they do not depend on any particular social welfare criterion. Each extension sheds new light on the role of central bank transparency. Focusing on the signaling role of the interest rate, we consider various degrees of transparency, ranging from full opacity, to just publishing the interest rate, to also revealing the signals and estimates of their precision. While uncertainty about the fundamentals results in the now familiar common knowledge effect, uncertainty about information precision creates a fog effect, which reduces the quality of decisions taken by the central bank and the private sector. In the absence of the fog effect, full transparency is generally not desirable, because it deprives the central bank from the ability to optimally manipulate private sector expectations. When the central bank’s fog is large, we find that full transparency is usually the best communication strategy. This result tends to survive when the private sector’s fog is large. Full opacity is only desirable when the central bank is poorly informed. Another result that emerges from our analysis is that it is usually desirable for the central bank to divulge some information, even if it is erroneous, and known to be erroneous. The reason is that, when the private sector knows that the central bank is mistaken, it needs to evaluate the extent of its mistakes.
Keywords: central bank transparency; monetary policy; information asymmetry;
JEL classification: E42, E52 E58