Monopoly with local knowledge of demand function

In this note, we propose a model where a quantity setting monopolist has incomplete knowledge of the demand function. In each period, the firm sets the quantity produced observing only the selling price and the slope of the demand curve at that quantity. Given this information and through a learning process the firm estimates a linear subjective demand curve. We show that the steady states of the dynamic equation are critical points of the objective profit function. Moreover, results depend on convexity/concavity of the demand. When the demand function is convex and the objective profit function has a unique critical point: the steady state is a globally stable maximum; conversely when then steady state is not unique, local maximums are locally stable, while local minimums are locally unstable. On the other hand when the demand function is concave, the unique critical point is a maximum: there can be stability or instability of the critical point and period two cycles around it via a flip bifurcation. Moreover, through simulations we can observe that, with a mixed inverse demand function, there are different dynamic behaviors, from stability to chaos and that we have transition to complex dynamics via a sequence of period-doubling bifurcations. Finally, we show that the same results can be obtained if the monopolist is a price setter.     

Hopf bifurcation and quasi-periodic dynamics in discrete multisector optimal growth models

This paper discusses the asymptotic stability of the steady state and the existence of a Hopf bifurcation in discrete time multisector optimal growth models. We obtain on the one hand a local turnpike theorem which guarantees the saddle point property for all discount rates. On the other hand, we provide a new proposition which gives some conditions ensuring local stability of the steady state if the impatience rate is not too high. A characterization of the boundδ*, above which the steady state is saddle-point stable, is also proposed in terms of indirect utility function's concavity properties. On this basis, some sufficient conditions for the existence of a Hopf bifurcation are stated. We thus prove the existence of quasi-periodic optimal paths in asymmetric models.     

On the asymptotic local stability of autonomous economic systems with nonnegativity conditions

A system of differential equations with nonnegativity conditions—a system of the type used to model price adjustment processes—is proved to be asymptotically locally stable if a subsystem without the nonnegativity conditions is asymptotically locally stable, provided a simple (mathematical) condition prevails.     

Gaussian estimation of a second order continuous time macroeconometric model of the UK

This paper describes the first application to an economy-wide macroeconometric model of recently developed methods for the exact Gaussian estimation of higher order continuous time dynamic models. The new model is formulated as a system of second order differential equations, thus providing a much richer dynamic specification than the predominantly first order continuous time macroeconometric models developed during the last 15 years. It also makes intensive use of economic theory to obtain a parsimonious parametrization, is designed in such a way as to permit a rigorous mathematical investigation of its steady state and asymptotic stability properties, and makes systematic use of the assumption of long-run rational expectations. In addition to the exact Gaussian estimates of the structural parameters, the paper includes the first set of continuous lag distributions derived from estimates that take account of the exact restrictions on the distribution of the discrete data implied by a continuous time model. It also includes the first estimates of the coefficient matrices of the exact discrete model, in its VARMAX form, satisfied by the discrete stock and flow data generated by a higher order continuous time dynamic model.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

Keynes-Friedman and Keynes-Marx models of monetary growth

The paper introduces a Keynes-Wicksell type model which generalizes a textbook model of Sargent (1987). It shows that the steady state of the model is locally asymptotically stable if two economically meaningful conditions are imposed on it. We then investigate two important limit cases with further stability properties. Case 1 has characteristics which are close to Sargent's Friedman version of such a Keynes-Wicksell dynamics, while case 2 is of a twofold Classical cross-dual type, with features of the process of equalizing profit rates as well as socalled profit-squeeze mechanisms. We close the paper with some numerical simulations of the latter approach to a Classical type of business cycle theory.     

Analysis on the dynamics of a Cournot investment game with bounded rationality

In this work, a dynamic system of investment game played by two firms with bounded rationality is proposed. It is assumed that each firm in any period makes a strategy for investment and uses local knowledge to make investment strategy according to the marginal profit observed in the previous period. Theoretic work is done on the existence of equilibrium solutions, the instability of the boundary equilibriums and the stability conditions of the interior equilibrium. Numerical simulations are used to provide experimented evidence for the complicated behaviors of the system evolution. It is observed that the equilibrium of the system can loose stability via flip bifurcation or Neimark–Sacher bifurcation and time-delayed feedback control can be used to stabilize the chaotic behaviors of the system.     

On the evolutionary selection of sets of Nash equilibria

It is well established for evolutionary dynamics in asymmetric games that a pure strategy combination is asymptotically stable if and only if it is a strict Nash equilibrium. We use an extension of the notion of a strict Nash equilibrium to sets of strategy combinations called ‘strict equilibrium set’ and show the following. For a large class of evolutionary dynamics, including all monotone regular selection dynamics, every asymptotically stable set of rest points that contains a pure strategy combination in each of its connected components is a strict equilibrium set. A converse statement holds for two-person games, for convex sets and for the standard replicator dynamic.     

Stability and bifurcation in a simple heterogeneous asset pricing model

Stability and bifurcation analysis of deterministic systems has been widely used in modeling financial markets. We develop a simple pricing model with two types of rational traders, fundamentalists and chartists, in order to study well price behavior in financial markets, we use student t distribution to replace traditional normal distribution to describe fundamental price process. We study the stability and bifurcation of the underling deterministic system and use numerical simulation to study the dynamic of the stochastic system, including autocorrelations structures and high kurtosis of the returns. It is found that the fundamental price becomes stable (unstable) when the activities from both types of traders are balanced (unbalanced).     

Out-of-equilibrium price dynamics

A trading-post organization of exchange is shown to determine an out-of-equilibrium price dynamics. The unique equilibrium of quasi-linear economies (defined by log-linear utility functions) is stable for the discrete time version of the dynamics. Equilibria that are stable for the continuous time version include those that satisfy the gross-substitutability property, the no-trade equilibria and, more generally, those for which trade intensity is relatively small. In addition, the set of stable equilibria is path-connected when endowments are allowed to vary without sign restrictions.      

STABILITY OF THE COURNOT SOLUTION AND FREE ENTRY: A NOTE*

Taking a traditional approach, I show that by using the discrete adjustment system, the Cournot solution in a free entry oligopoly with increasing returns to scale is stable only when the number of firms is very small. We must pay attention to stability when using the free entry Cournot oligopoly model.     

A note on the sustainability of primary budget deficits

This note studies the sustainability of primary budget deficits in a situation where the public sector has also incurred debt to the private sector. It shows that if the rate of pure time preference is small relative to the population growth rate, if public consumption constitutes a modest fraction of output, and if the primary budget deficit is sufficiently small, then it is also sustainable, and the associated steady state is locally stable.     

From evolutionary to strategic stability

A connected component of Nash equilibria is (dynamically) potentially stable if there exits an evolutionary selection dynamics from a broad class for which the component is asymptotically stable. A necessary condition for potential stability is that the component's index agrees with its Euler characteristic. Second, if the latter is nonzero, the component contains a strategically stable set. If the Euler characteristic would be zero, the dynamics (that justifies potential stability) could be slightly perturbed so as to remove all zeros close to the component. Hence, any robustly potentially stable component contains equilibria that satisfy the strongest rationalistic refinement criteria.     

Inflation and distribution in a Keynes-Wicksell model of the business cycle

The paper integrates a Kaleckian hypothesis of markup pricing and markup adjustments into a dynamic IS-LM model of Keynes-Wicksell type. Local stability and instability are characterized by different speeds of adjustments in the adaptive expectations of inflation. Application of the Hopf bifurcation theorem proves existence of periodic orbits in the three-dimensional state space. The global dynamics is investigated by means of computer simulations. A suitable modification of the rule of adaptive expectations checks the explosive tendencies and gives rise to an apparently unique and stable limit cycle. It is sketched how the evolution of the most important variables over this growth cycle compares to empirical stylized facts.     

Tolerance and Herd Behavior: Sophistication of Ideas under Payoff Complementarity

In this paper, I first show that there may be inefficient herd behavior even with a continuous choice set and a continuous loss function if there is payoff complementarity. Then, I show that the probability of inefficient herd behavior is asymptotically zero and the choice sequence converges to the optimal one almost surely if people have even a small amount of tolerance that they are willing to accept one another as a partner in joint work. This result is closely related to the argument in J. S. Mill’s On Liberty, where he states that tolerance for others’ ideas is essential for sophistication of ideas.     

The local bifurcation of Ramsey equilibrium

Summary The elasticity of substitution has been proposed as one factor in the generation of aggregate fluctuations in dynamic models with incomplete markets. We study the existence of periodic solutions in a one-sector neoclassical capital accumulation model under borrowing constraints with infinitely-lived heterogeneous agents. A dynamical system representing an equilibrium profile with only the most patient agent holding capital is analyzed when capital income is not an increasing function of total capital. Conditions for the linear approximation system at a steady state to have an eigenvalue of — 1 are found. A one-parameter family of maps based on a perturbation of the production function is introduced and the dynamical system is reduced to 1 dimension via an application of a center manifold theorem. Conditions for a stable flip bifurcation are shown to hold at the steady state.This paper is dedicated to Professor Nicolas Spulber, one of the great pioneers in the study of growth theory. He brought us together and encouraged our joint work on Ramsey equilibrium theory.We thank Nicolas Spulber for his usual insightful comments and we thank Michele Boldrin for a useful discussion. We also thank a referee for suggesting improvements in the paper as well as the seminar participants at the University of Rochester and Cornell University for their comments on the paper.     

Technology gap and international trade: an evolutionary model

We propose a model of the international technology gap that focuses on two sources of self-reinforcing mechanisms in the industrial competition: (i) a positive feedback that runs from innovations to profits to R & D expenditures, and (ii) learning effects in R & D and in production. We find that, if the cost of labor is lower in the late-starter country, several dynamic paths are possible, including one in which the late-starter catches up and then reverses the technology gap. When international diffusion of technology is introduced, the system has a bifurcation structure: if technology diffusion is relatively slow, there are two steady-state levels of the technology gap, one in favor of each country; if diffusion is fast, there is a unique stable equilibrium gap in favor of the country that has an exogenous (Ricardian) cost advantage.     

The Effects of Delays on the Stability and Rate of Convergence to Equilibrium of Oligopolies*

The stability of oligopolies is investigated using the Cournot strategy and a general output adjustment procedure. Information delays are incorporated in the adjustment procedure and their specific effects upon the dynamic stability of oligopolies is investigated. The stability of equilibrium for discrete models is unaffected by information delays and remoteness of planning horizons; however the rate of convergence to equilibrium may be slowed. For continuous models with delays, some restraints are required for stability.     

Differences in measures of the fiscal multiplier and the reduced-form vector autoregression

The literature has recently asked whether the effects of fiscal policy vary with the state of the economy (Christiano, Eichenbaum, and Rebelo 2011; Rendahl 2014; Auerbach and Gorodnichenko 2012). We study this question in the context of vector autoregression (VAR) estimation. We show formally that, if (asymptotically) the parameters of the reduced-form VAR differ, then the dynamic effects of fiscal policy differ as well, generically and for any set of identification assumptions. Thus, in theory, the econometrician can detect these differences (either across time or space) generically just by relying on reduced-form VAR estimation.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.