Paper: arXiv 2305.00541

Authors: René Aid, Matteo Basei, Giorgio Ferrari

Abstract

We consider a mean-field model of firms competing à la Cournot on a commodity market, where the commodity price is given in terms of a power inverse demand function of the industry-aggregate production. Investment is irreversible and production capacity depreciates at a constant rate. Production is subject to Gaussian productivity shocks, while large non-anticipated macroeconomic events driven by a two-state continuous-time Markov chain can change the volatility of the shocks, as well as the price function. Firms wish to maximize expected discounted revenues of production, net of investment and operational costs. Investment decisions are based on the long-run stationary price of the commodity. We prove existence, uniqueness and characterization of the stationary mean-field equilibrium of the model. The equilibrium investment strategy is of barrier-type and it is triggered by a couple of endogenously determined investment thresholds, one per state of the economy. We provide a quasi-closed form expression of the stationary density of the state and we show that our model can produce Pareto distribution of firms’ size. This is a feature that is consistent both with observations at the aggregate level of industries and at the level of a particular industry. We establish a relation between economic instability and market concentration and we show how macroeconomic instability can harm firms’ profitability more than productivity fluctuations.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical mean-field game framework with advanced stochastic controls and Markov chain regime-switching. While it provides insightful analytical results regarding firm size distribution and market concentration, it lacks empirical validation and backtesting with real-world market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Model Irreversible Investment & Cournot Competition in a Two-Regime Economy"] --> B["Inputs: Markov Macro-States, Gaussian Shocks, Power Inverse Demand, Capacity Depreciation"]
    B --> C["Methodology: Stationary Mean-Field Equilibrium MFG & Optimal Stopping"]
    C --> D["Analytical Process: Derive HJB Equation, Free-Boundary Problem & Kolmogorov Equation"]
    D --> E["Outcome 1: Unique Stationary MFE with Two State-Dependent Investment Thresholds"]
    D --> F["Outcome 2: Quasi-Closed Form Density Yielding Pareto Firm-Size Distribution"]
    D --> G["Outcome 3: Macro Instability Harms Profitability More Than Productivity Shocks"]