arXiv subjects

Robert M. Anderson

Publications and source records attributed to Robert M. Anderson.

7 recordsLinked to original sources

arXiv2026

Existence of Equilibria in Large Competitive Markets with Bads, Production and Comprehensive Externalities

This paper establishes existence of equilibrium in a measure-theoretic general equilibrium (MGE) model with production, bads, and comprehensive externalities. These features are jointly essential for modeling perfect competition in which emissions of production byproducts impose harm on agents. We show that, when bads and externalities are modeled in an economically natural way, equilibrium exists. This yields the first existence theorem with bads for MGE models, the benchmark for perfect competition, overcoming Hara (2005)'s nonexistence example. The proof uses nonstandard analysis, which provides a systematic technique to extend results for finite to infinite models.

econ.TH↗

arXiv2023

General Equilibrium Theory for Climate Change

We propose two general equilibrium models, quota equilibrium and emission tax equilibrium. The government specifies quotas or taxes on emissions, then refrains from further action. Quota equilibrium exists; the allocation of emission property rights strongly impacts the distribution of welfare. If the only externality arises from total net emissions, quota equilibrium is constrained Pareto Optimal. Every quota equilibrium can be realized as an emission tax equilibrium and vice versa. However, for certain tax rates, emission tax equilibrium may not exist, or may exhibit high multiplicity. Full Pareto Optimality of quota equilibrium can often be achieved by setting the right quota.

econ.TH↗

arXiv2023

On Existence of Berk-Nash Equilibria in Misspecified Markov Decision Processes with Infinite Spaces

Model misspecification is a critical issue in many areas of theoretical and empirical economics. In the specific context of misspecified Markov Decision Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium and established its existence in the setting of finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo, as well as Gaussian and log-normal distributions, and CARA, CRRA and mean-variance preferences) involve continuous state or action spaces, and are thus not covered by the Esponda-Pouzo existence theorem. We extend the existence of Berk-Nash equilibrium to compact action spaces and sigma-compact state spaces, with possibly unbounded payoff functions. A complication arises because the Berk-Nash equilibrium notion depends critically on Radon-Nikodym derivatives, which are necessarily bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis and, relative to previous applications of nonstandard analysis in economic theory, draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

econ.TH↗

arXiv2020

Nonstandard Representation of the Dirichlet Form

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well-approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we prove a generalization of a well-known comparison theorem for Markov chains on finite state spaces, and also relate our results to previous generalization attempts.

math.PR↗

arXiv2020

Loeb Extension and Loeb Equivalence

In Keisler and Sun (2004), the authors raise several open problems on Loeb equivalences between various internal probability spaces. We provide counter-examples for the first two open problems. Moreover, we reduce the third open problem to the following question: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of the Loeb extension of any one of the internal algebra?

math.LO↗

arXiv2020

Drift, Minorization, and Hitting Times

The "drift-and-minorization" method, introduced and popularized in (Rosenthal, 1995; Meyn and Tweedie, 1994; Meyn and Tweedie, 2012), remains the most popular approach for bounding the convergence rates of Markov chains used in statistical computation. This approach requires estimates of two quantities: the rate at which a single copy of the Markov chain "drifts" towards a fixed "small set", and a "minorization condition" which gives the worst-case time for two Markov chains started within the small set to couple with moderately large probability. In this paper, we build on (Oliveira, 2012; Peres and Sousi, 2015) and our work (Anderson, Duanmu, Smith, 2019a; Anderson, Duanmu, Smith, 2019b) to replace the "minorization condition" with an alternative "hitting condition" that is stated in terms of only one Markov chain, and illustrate how this can be used to obtain similar bounds that can be easier to use.

math.PR↗

arXiv2019

Mixing Times and Hitting Times for General Markov Processes

The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metroplis-Hastings chains, even though they usually do not satisfy the strong Feller property.

math.PR↗