Competition versus Cooperation: A class of solvable mean field impulse control problems

TL;DR

This paper analyzes a class of solvable mean field impulse control problems with economic applications, extending classical models to include state-dependent prices and comparing competitive and cooperative scenarios.

Contribution

It introduces explicit solutions for mean field impulse control problems with state-dependent prices and compares equilibrium and cooperative strategies.

Findings

Existence of explicit threshold-type equilibrium strategies.

Characterization of optimal thresholds in both competitive and cooperative settings.

Illustrative example demonstrating the theoretical results.

Abstract

We discuss a class of explicitly solvable mean field type control problems/mean field games with a clear economic interpretation. More precisely, we consider long term average impulse control problems with underlying general one-dimensional diffusion processes motivated by optimal harvesting problems in natural resource management. We extend the classical stochastic Faustmann models by allowing the prices to depend on the state of the market using a mean field structure. In a competitive market model, we prove that, under natural conditions, there exists an equilibrium strategy of threshold-type and furthermore characterize the threshold explicitly. If the agents cooperate with each other, we are faced with the mean field type control problem. Using a Lagrange-type argument, we prove that the optimizer of this non-standard impulse control problem is of threshold-type as well and characterize the optimal threshold. Furthermore, we compare the solutions and illustrate the findings in an example.