Discrete Choices under Social Influence: Generic Properties

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP). Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When j, the ratio of the social influence strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of j above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria. Beyond this first result, we exhibit the generic properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend only on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite). The main results are summarized as phase diagrams in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.Comment: 42 pages, 15 figure

A Framework for Studying Economic Interactions (with applications to corruption and business cycles)

Most economic models implicitly or explicitly assume that interactions between economic agents are 'global' - in other words, each agent interacts in a uniform manner with every other agent. However, localized interactions between microeconomic agents are a pervasive feature of reality. What are the implications of more limited interaction? One set of mathematical tools which appears useful in exploring the economic implications of local interactions is the theory of interacting particle systems. Unfortunately, the extant theory mainly addresses the long-time behavior of infinite systems, and focuses on the issue of ergodicity; many economic applications involve a finite number of agents and are concerned with other issues, such as the extent of shock amplification. In this paper, I introduce a framework for studying local interactions that is applicable to a wide class of games. In this framework, agents receive shocks which are stochastically independent; payoffs depend both upon the shocks and the strategies of other agents. In finite games, ergodicity is straightforward to determine. In finite games which evolve in continuous time, the stationary distribution (if it exists) may be computed easily; furthermore, in this class of games, I prove that any stationary distribution may be attained by suitable choice of payoff functions using shocks which are distributed uniform on (0, 1). In systems in which all interactions are global, I prove that nonlinear behavior can arise even in the infinite limit (thus demonstrating that laws of large numbers can fail in systems characterized by interaction), despite the fact that the only driving forces are agent-level iid disturbances. Using numerical methods, I investigate the properties of the processes as one passes from discrete to continuous time, as one alters the pattern of interaction, and as one increases the number of interacting agents. In so doing, I provide further evidence that the existence of local interactions can change the aggregate behavior of an economic system in fundamental ways, and that the form of that interaction has important implications for its dynamic properties.

Discrete Choices under Social Influence: Generic Properties

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When $j$, the ratio of the social influene strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of $j$ above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.Beyond this first result, we exhibit the {\em generic} properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend {\em only} on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite).The main results are summarized as {\em phase diagrams} in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.discrete choice; social influence; externalities; heterogeneous agents; socioeconomic behavior

The Futility of Utility: how market dynamics marginalize Adam Smith

Econometrics is based on the nonempiric notion of utility. Prices, dynamics, and market equilibria are supposed to be derived from utility. Utility is usually treated by economists as a price potential, other times utility rates are treated as Lagrangians. Assumptions of integrability of Lagrangians and dynamics are implicitly and uncritically made. In particular, economists assume that price is the gradient of utility in equilibrium, but I show that price as the gradient of utility is an integrability condition for the Hamiltonian dynamics of an optimization problem in econometric control theory. One consequence is that, in a nonintegrable dynamical system, price cannot be expressed as a function of demand or supply variables. Another consequence is that utility maximization does not describe equiulibrium. I point out that the maximization of Gibbs entropy would describe equilibrium, if equilibrium could be achieved, but equilibrium does not describe real markets. To emphasize the inconsistency of the economists' notion of 'equilibrium', I discuss both deterministic and stochastic dynamics of excess demand and observe that Adam Smith's stabilizing hand is not to be found either in deterministic or stochastic dynamical models of markets, nor in the observed motions of asset prices. Evidence for stability of prices of assets in free markets simply has not been found.Comment: 46 pages. accepte

Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach

We analyze the Glauber dynamics for a bi-populated Curie-Weiss model. We obtain the limiting behavior of the empirical averages in the limit of infinitely many particles. We then characterize the phase space of the model in absence of magnetic field and we show that several phase transitions in the inter-groups interaction strength occur.Comment: 18 pages, 3 figure