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Nirei and Scheinkman (2021) proposed an equilibrium model of price adjustments with menu-costs with a finite number of firms and derived a “reproduction number” for repricing and a limit functional form for the distribution of the number of simultaneously price-adjusting firms. We show that the distribution of price-changes in data from the Billion Prices Project is well fitted by this functional form and exhibits a reproduction number that is close to unity, indicating that complementarity in price-changes plays a major role in repricings.

In this Supplementary Appendix, we extend the baseline model of "Repricing Avalanches" to the case where productivity is not constant. Even though some portions are straightforward extension of the baseline model, we allow for some overlap with the main text for the sake of a self-contained exposition of the extended model.

We study the speculative value of a finitely lived asset when investors disagree and short sales are limited. In this case, investors are willing to pay a speculative value for the resale option they obtain when they acquire the asset. Using martingale arguments, we characterize the equilibrium speculative value as a solution to a fixed point problem for a monotone operator $$\mathbb F$$ . A Dynamic Programming Principle applies and is used to show that the minimal solution to the fixed-point problem is a viscosity solution of a naturally associated (non-local) obstacle problem. Combining the monotonicity of the operator $${\mathbb {F}}$$ and a comparison principle for viscosity solutions to the obstacle problem we obtain several comparison of solution results. We also use a characterization of the exercise boundary of the obstacle problem to study the effect of an increase in the costs of transactions on the value of the bubble and on the volume of trade, and in particular to quantify the effect of a small transaction (Tobin) tax.

Recently, J. Scheinkman and B. Le Baron have analyzed a time series of weekly stock returns from the point of view of nonlinear dynamics. In particular, using the Grassberger-Procaccia algorithm, they found the data to be compatible with a deterministic dynamical system of low dimension perturbed by a moderate amount of noise. This chapter analyses the same time series, and obtain an estimate of Lyapunov exponents, that is, rates of exponential growth of small perturbations to initial conditions. It also analyzes the data for temporal homogeneity by producing a recurrence plot to obtain this plot the weekly returns time series is first used to generate a six-dimensional trajectory by the time-delay method. The recurrence plot, apart form a short exceptional period near the beginning of the time series, appears reasonably uniform. This seems to indicate that there is not much drift–in the more than twenty years studied–for the behavior of weekly returns over a few weeks.

In this paper, we establish existence and uniqueness results for equilibria in systems with an infinite number of agents and with local and global social interactions. We also examine the structure of the equilibrium distribution and derive a “Markov”property for the equilibrium distribution of a class of spatially homogeneous systems.

We consider an evolution non-local free boundary problem that arises in the modelling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show that the solution is convex in space, and establish several monotonicity properties of the solution and of the free boundary with respect to parameters of the problem. To study the free boundary, we use, in particular, the fact that the odd part of the solution solves a more standard obstacle problem. We show that the free boundary is and describe the asymptotics of the free boundary as c , the cost of transacting the asset, goes to zero.

This paper shows how to identify nonparametrically scalar stationary diffusions from discrete-time data. The local evolution of the diffusion is characterized by a drift and diffusion coefficient along with the specification of boundary behavior. We recover this local evolution from two objects that can be inferred directly from discrete-time data: the stationary density and a conveniently chosen eigenvalue–eigenfunction pair of the conditional expectation operator over a unit interval of time. This construction also lends itself to a spectral characterization of the over-identifying restrictions implied by a scalar diffusion model of a discrete-time Markov process.

This paper presents a test of independence that can be applied to the estimated residuals of any time series model that can be transformed into a model driven by independent and identically distributed errors. The first order asymptotic distribution of the test statistic is independent of estimation error provided that the parameters of the model under test can be estimated -consistently. Because of this, our method can be used as a model selection tool and as a specification test. Widely used software1 written by Dechert and LeBaron can be used to implement the test. Also, this software is fast enough that the null distribution of our test statistic can be estimated with bootstrap methods. Our method can be viewed as a nonlinear analog of the Box-Pierce Q statistic used in ARIMA analysis.

In this paper, we consider an evolution non local free boundary problem that arises in the modeling of speculative bubbles. The solution of the model is the speculative component in the price of an asset. In the framework of viscosity solutions, we show the existence and uniqueness of the solution. We also show the convexity in space of the solution, and various monotonicity properties of the solution and of the free boundary with respect to parameters of the problem.In order to study the free boundary, we use in particular the fact that the odd part of the solution solves a more standard obstacle problem. We show moreover that the free boundary is Lipschitz continuous, and describe the asymptotics of the free boundary as c, the cost of transacting the asset, goes to zero.

Recursive utility models of the type introduced by Kreps and Porteus (1978) are used extensively in applied research in macroeconomics and asset pricing in environments with uncertainty. These models represent preferences as the solution to a nonlinear forward-looking difference equation with a terminal condition. Such preferences feature investor concerns about the intertemporal composition of risk. In this paper we study infinite horizon specifications of this difference equation in the context of a Markov environment. We establish a connection between the solution to this equation and to an arguably simpler Perron-Frobenius eigenvalue equation of the type that occurs in the study of large deviations for Markov processes. By exploiting this connection, we establish existence and uniqueness results. Moreover, we explore a substantive link between large deviation bounds for tail events for stochastic consumption growth and preferences induced by recursive utility.

Publisher Summary This chapter surveys a set of mathematical and statistical tools that are valuable in understanding and characterizing nonlinear Markov processes. Such processes are used extensively as building blocks in economics and finance. Operator methods begin with a local characterization of the Markov process dynamics. This local specification takes the form of an infinitesimal generator. The infinitesimal generator is itself an operator mapping test functions into other functions. From the infinitesimal generator, one constructs a family (semigroup) of conditional expectation operators. The operators exploit the time-invariant Markov structure. Each operator in this family is indexed by the forecast horizon; the interval of time between the information set used for prediction and the object that is being predicted. Operator methods allow ascertaining global and, in particular, long-running implications from the local or infinitesimal evolution. These global implications are reflected in the implied stationary distribution; the analysis of the eigenfunctions of the generator that dominate in the long run; and the construction of likelihood expansions and other estimating equations.

We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low frequency data. We also explore implications for stationary, possibly non-reversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretely-sampled data.

We investigate a method for extracting nonlinear principal components. These principal components maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these principal components. We also characterize the limiting behavior of the associated eigenvalues, the objects used to quantify the incremental importance of the principal components. By exploiting the theory of continuous-time, reversible Markov processes, we give a different interpretation of the principal components and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the principal components maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the principal components behave as scalar autoregressions with heteroskedastic innovations. Finally, we explore implications for a more general class of stationary, multivariate diffusion processes.

We study the existence of solutions to a nonlinear functional differential system that describes the equilibrium of a dynamic stochastic economy with heterogeneous agents facing borowing constraints. Our model is a generalization of the one studied in Scheinkman and Weiss [3] to display the effect of borrowing constraints in aggregate economic activity, and the mathematical techniques that we developp can be useful to deal with more complex models. The system has aspects of a free boundary value problem in which there are different equations for different domains, with the domains themselves and their boundary determined by the solutions. It also has some of the characteristics of hyperbolic systems that one would obtain by differentiating Hamilton-Jacobi equations. These aspects combine to require, at least apparently, a very specific new approach to the existence problem. One payoff of this technique is that the existence proof is also an algorithm for computation.

We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return trade-off.

In this paper, we establish a convergence result for equilibria in systems of social interactions with many locally and globally interacting players. Assuming spacial homogeneity and that interactions between dierent agents are not too strong, we show that equilibria of systems with finitely many players converge to the unique equilibrium of a benchmark system with infinitely many agents. We prove convergence of individual actions and of average behavior. Our results also apply to a class of interaction games (Morris 1997).

This paper treats dynamic optimization problems from the point of view of programming in infinite dimensional spaces. Sufficient conditions (of a dominant diagonal nature) for smooth dependence of optimal paths on initial conditions and other parameters are given. It is also shown that the smooth dependence and the so called "turnpike property" are very closely related. This relationship is used to prove a turnpike theorem for a model that may be interpreted as a multisectoral model of optimal growth with positive discounting and to show that the turnpike property is kept under "small" perturbations.

Necessary conditions for optimality in infinite horizon concave maximization problems, which include a generalized “Euler equation” and a transversality condition at infinity, are established. The formulation admits implicit state constraints and utility functions with infinite steepness at the boundary. For these reasons support prices present discontinuities and a theorem characterizing such discontinuities is presented.

This paper considers a class of discrete-time, infinite-horizon optimization problems arising in economics. Necessary optimality conditions for such problems consist of an Euler equation and a transversality condition at infinity. Our concern here is with the latter condition. Previous results for this type of problem have required such restrictions as concavity and boundedness of the utility functions whose sum is being maximized. Here we derive a transversality condition that remains valid under considerably more general hypotheses. In doing so, we introduce methods that may also be applicable to other situations, such as optimization in continuous time.

This paper contains results on local and global stability of n-sector growth models when utility is discounted mostly for small rates of discount. It is well known that when future utility is not discounted one can prove precise results about optimal steady states (OSS's) under fairly general assumptions. In particular, existence, uniqueness, and turnpike properties have been established by several authors. The counter examples presented by Kurz, Sutherland, and Weitzman, however, show that when utility is discounted, additional assumptions are required to obtain turnpike results. In general, it would be interesting to know how the submanifolds of stability change as δ changes. One hopes that certain conditions on the utility function would be sufficient to “classify” the submanifolds of stability and instability. Such a question is apparently very difficult to answer, but we think that the results obtained here will help in this task. The proof that the turnpike theorem holds for discount factors near one is divided in two parts. First, we prove that optimal paths “visit” neighborhoods of the modified OSS's. Then, we prove that local stability holds for such neighborhoods. In order to show this fact, we must prove that the local “stable manifold” varies continuously with the discount factor. This roundabout method is necessary since our problem is similar to proving uniform continuity with respect to a parameter of solutions of a differential equation in a noncompact interval of time. Other problems analyzed here include uniqueness and continuity of OSS's. We also discuss the relation between the saddle-point property and the local stability of infinite horizon optimal growth paths.

Publisher Summary This chapter provides an overview of comparative dynamics. It discusses the model used in Araujo and Scheinkman. The chapter provides proof of a general negative-definiteness property of stable models. It also presents a sharper version of the dominant diagonal assumption used in Araujo and Scheinkman to produce more precise comparative dynamics results. The chapter shows the application of the results to an adjustment cost model due to Lucas. In the general equilibrium theory, the dominant diagonal assumption was frequently used to allow one to sign the impact of a change of a parameter in the solution. The chapter highlights the way by which the dominant diagonal, coupled with restriction on the sign of elements of the matrix, can give comparative dynamics information.

Publisher Summary This chapter discusses the global asymptotic stability of optimal control systems with applications to the theory of economic growth. The qualitative study of optimal economic growth has attracted the attention of economic theorists for a few years. One major focus of this research has been to find sufficient conditions on models of economic growth for the convergence of growth paths to a steady state. In this symposium volume of the Journal of Economic Theory, Cass and Shell present the Hamiltonian formulation of competitive dynamical systems that arise in capital theory. The chapter presents a set of sufficient conditions on the Hamiltonian for such dynamical systems to converge to a steady state as time tends to infinity. The chapter also presents new results that build on the work of Cass and Shell, Rockafellar, and Hartman and Olech. It discusses the three basic types of results obtained and their relation to the literature.